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\title{Generalized norms in groups}%Paper title
\author{Fedir M. Lyman, Tenyana D. Lukashova, Maryna G. Drushlyak}%Authors names
\shorttitle{Generalized norms in groups}%Short paper title for the headers
\shortauthor{F. Lyman, T. Lukashova, M. Drushlyak}%Short authors names for the headers
\address [F. Lyman]{Romenska Str.~87, 40002 Sumy, Ukraine}{mathematicsspu@mail.ru}{}%www.fauthor.domain.com
\address [T. Lukashova]{Romenska Str.~87, 40002 Sumy, Ukraine}{tanya.lukashova2015@gmail.com}{}
\address [M. Drushlyak]{Romenska Str.~87, 40002 Sumy, Ukraine}{marydru@mail.ru}{}

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\date{\today}

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\begin{abstract}
In this survey paper the authors have tried to specify all the known results
concerning the norms of the group and their generalizations. Special attention is
paid to the analysis of their own research of different generalized norms,
particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic
subgroups, the norm of infinite subgroups, the norm of infinite Abelian
subgroups and the norm of other systems of Abelian subgroups.

\end{abstract}

\subjclass{2001}{20D25; 20E28}

\keywords{norm of group, generalized norm, Wielandt subgroup, norm of
non-cyclic subgroups, norm of Abelian non-cyclic subgroups, norm of infinite
subgroups, norm of infinite Abelian subgroups, norm of decomposable subgroups}

\section*{Introduction}
In group theory results concerning the study of characteristic subgroups (in
particular, the center, the derived subgroup, Frattini subgroup, etc.) and the
impact of properties of these subgroups on the structure of the group are hold
a special place. Nowadays the list of such haracteristic subgroups can be
recharged through a different $\Sigma $-norms of a group.

Let $\Sigma $ be the system of all subgroups of the group which have some
theoretical group property. For example, $\Sigma $ can consist of all subgroups
of the group, of all cyclic, all non-cyclic, all Abelian, all non-Abelian, all
subnormal, all maximal, all infinite subgroups of the group. $\Sigma $-norm of
a group G is called the intersection $N_{\Sigma } (G)$ of the normalizers of
all subgroups of the group which belong to the system $\Sigma $. In the case
$\Sigma $=$\emptyset$ we assume that \textit{G=}$N_{\Sigma } (G)$.

By the definition of the $\Sigma $-norm it follows that it is a characteristic
subgroup of the group and contains the center of the group. Also, $N_{\Sigma }
(G)$ is the maximal subgroup of the group that normalizes all $\Sigma
$-subgroups of the group. Therefore, all subgroups of $\Sigma $-norm, which
belong to the system $\Sigma $, are normal in $N_{\Sigma}(G)$ (although these
subgroups may not exist).

Considering the $\Sigma $-norm, there are several problems associated with the
study of the properties of the group with the given system $\Sigma $ of
subgroups and some restrictions, which the norm satisfies. Some tasks of this
type were solved by many algebraists depending on the choice of a system
$\Sigma $ and properties of the $\Sigma $-norm.

Knowing the structure of $\Sigma$-norms and the nature of its attachment to the
group, the properties of the group can be characterized in many cases. For
example, if the $\Sigma $-norm coincides with the group and $\Sigma \ne
\emptyset$, then all subgroups of the system $\Sigma $ are normal in the group.
First non-Abelian groups with this property were considered in XIX century by
R.~Dedekind \cite{Ded:1}, who gave a complete description of finite non-Abelian
groups, all subgroups of which are normal, and called them Hamiltonian groups.
Infinite Hamiltonian groups were described in 1933 by R.~Baer \cite{Baer:1}. A
union of sets of Abelian and Hamiltonian groups are called the set of Dedekind
groups.

However, the study of groups with other systems $\Sigma $ of normal subgroups
were continued only in the second part of XX century, that slowed down the
study of $\Sigma $-norms. Research of S.~M.~Chernikov and his disciples are
research of this kind. Thus nowadays the structure of groups that coinside with
the norm $N_{\Sigma}(G)$ is known for many systems of subgroup. So the question
on the study of the properties of groups, in which the $\Sigma $-norm is a
proper subgroup, arises naturally.

\section{The norm of group and subgroups closed to it}

For the first time the problem of the study of the properties of groups, which
differ from the $\Sigma $-norm, was formulated by R.~Baer in 30 years of the
last century. In \cite{Baer:2}  he was introduced a subgroup $N(G)$, which is
the intersection of normalizers of all subgroups of a group, and called it the
norm of the group \textit{G}. It is clear that the norm $N(G)$ is the $\Sigma
$-norm of the group for the system $\Sigma $, which consists of all subgroups
of the group. The norm $N(G)$ is contained in all other $\Sigma $-norms and
they can be considered as its generalizations. It is also clear that Dedekind
groups coincide with their norms, so the index of the norm in a group can serve
as a certain "degree of Dedekindness'' of a group.

The norm of group was studied by R.~Baer [\ref{Baer:2}--\ref{Baer:9}] and
several other authors [\ref{Wos:1}--\ref{Rus:2}]. R.~Baer noticed that the
restrictions that are imposed on the norm of the group impact on the structure
of the group in a certain way. In particular, there is a proposition.

\begin{proposition}\label{p:1.1} (\cite{Baer:2}) If a norm $N(G)$ of a group $G$ is Hamiltonian, then the following
propositions take place:

\textit{1)} $G$ is a periodic group;

\textit{2)} $G$ contains no elements which orders are divisible by 8;

\textit{3)} all elements of a group $G$, which have orders multiple of 4, can
be represented in the form of $za$, where $a\in N\left(G\right)$,
$\left|a\right|=4$, and $z\in G$, more over $z$ is permutable with each element
of norm $N(G)$;

\textit{4)} any element, which order is not divisible by 4, is permutable with
every element of the subgroup $N(G)$.
\end{proposition}


Studying the relations between the norm and the center of the group R.~Baer
showed that the norm coincides with the center of the group, if it contains
elements of infinite order \cite{Baer:2}. Another important result, which characterizes the impact of the center of the group on its norm, was established in \cite{Baer:9}.

\begin{proposition}\label{p:1.2} (\cite{Baer:9}) Norm of a group is trivial if and only if G is a group with trivial center.
\end{proposition}


Continuing the study of the properties of the norm of a group L.~Wos \cite{Wos:1} established that the norm $N(G)$ is contained in the third hypercentre of the group, and the group of automorphisms, which are induced on the subgroup \textit{N(G)} by $G$, is nilpotent of class at most 2. In addition, it was proved that the norm of the group is contained in the second hypercentre if and only if the group of automorphisms induced on $N(G)$ by the group $G$ is Abelian. This result was substantially refined by E.~Shenkman in \cite{Schen:1}.

\begin{proposition}\label{p:1.3} (\cite{Schen:1}) Norm N(G) of a group G is contained in the second hypercentre of $G$. Derived subgroup $G'$  is a subgroup of centralizer of a norm $N(G)$ in $G$.
\end{proposition}


So the group of automorphisms induced on $N(G)$ by $G$ is Abelian. Let's note that $N(G)=E$ in groups with trivial center by Proposition \ref{p:1.3}.


In \cite{Baer:5, Baer:7} the properties of periodic groups with an Abelian norm
quotient group were considered. In particular, in \cite{Baer:5} it was proved
that a periodic group $G$, which quotient group $G/N(G)$ is Abelian and
$N(G)\neq Z(G)$, is a direct product of its primary components, and its norm
$N(G)$ is a direct product of norms of these components.


In this regard, let's note that unlike some other characteristic subgroups (the
center of the group, the derived subgroup, Fitting subgroup and others) the
norm of the direct product of arbitrary subgroups is not equal to the direct
product of the norms of the respective components in general case.


\begin{example}\label{e:1.1} Let $G=Q\times B$, where $B$ is a non-periodic Abelian group of rank 1, $Q$ is the quaternion group of order 8. In this group

\centerline {$N(G)=N(Q\times B)=Q^{2} \times B\ne N(Q)\times N(B)=G$.}
\end{example}

The problem of finding the norms of direct products of groups was studied by J.~Evan in \cite{Evan:1}.

Let's note also the following result of R.~Baer, which characterizes the
properties of $p$-groups with Abelian norm quotient group.

\begin{proposition}\label{p:1.4}(\cite{Baer:7}) If $G$ is p-group $\left(p\ne 2, p\ne 3\right)$ that has an Abelian quotient group for the norm $N(G)$, more over $N\left(G\right)\ne Z\left(G\right)$ and $p^{r}$ is the exponent of the group $C_{G}(N(G))$, then:

\textit{1)} $G$ is a group of finite exponent;

\textit{2)} $N(G)/Z(G)$ is a cyclic group, the order of which is equal to the
exponent of the group of automorphisms induced on $N(G)$ by $G$;

\textit{3)} centralizer $C_{G}(N(G))$ consists of those and only those elements $x\in G$  for which $x^{p^{r}} =1$.
\end{proposition}


The restrictions $p\ne 2,p\ne 3$ in proposition 1.4 are significant, as
evidenced by the examples of the respective groups (see \cite{Baer:5}).


Recently the interest to the norm $N(G)$ of group is not reduced, as
evidenced by a number of researches [\ref{Evan:1}--\ref{Mil:1}], devoted to the study
of its properties. In particular, in \cite{Bryce:4, Bryce:5} R.~Bryce and
J.~Cossey considered series of norms

$$1=N_{0}(G)\subseteq N_{1}(G)\subseteq ...\subseteq N_{i}
(G)\subseteq ...,$$
where $N_{i} (G)/N_{i-1} (G)=N(G/N_{i-1}(G))$ for $i\ge 1$.

It was proved that in the class of 2-groups from the fact that quotient group
$N_{i+1} (G)/N_{i} (G)$ is Hamiltonian it follows that $N_{i+1} (G)=G$.
Moreover a finite 2-group, in which the quotient group $G/N(G)$ is Hamiltonian,
but any quotient group $N_{i}(G)/N_{i-1}(G)$ is not Hamiltonian, has order
$2^{7}$ and is uniquely determined up to isomorphism \cite{Bryce:5}.

Starting from R.~Baer, L.~Wos and E.~Schenkman research of the norm $N(G)$ was
concerned its relations with the centre of the group. In particular, in
\cite{Beidl:1} J.~Beidleman, H.~Heineken and M.~Newell have shown that in an
arbitrary $p$-group $G$ either quotient group $G/Z(G)$ or group
$\left[G,N(G)\right]$ is cyclic. In this article the problem of the impact of
properties of a norm of a group and its center on the capability of a group $G$ is
considered.

A group $G$ is called capable, if it is a group of inner automorphisms of some
group $H$ that is $G\cong H/Z(H)$. For the first time R.~Baer \cite{Baer:10}
studied such groups. He described capable finitely generated Abelian groups
\\
\centerline{$G=Z_{n_{1} } \oplus Z_{n_{2} } \oplus ...\oplus Z_{n_{k}}$,}
\\
where $n_{i+1}\vdots n_{i} ,{\rm \; }n_{i} \in N\bigcup \left\{0\right\}$ and $Z_{n_{i}} =Z$
is an infinite cyclic group for $n_{i} =0$. It was found that the group $G$ is
capable if and only if $k\ge 2$ and $n_{k-1} =n_{k}$. The Baer's
characterization remains the only full and complete for a certain class of
capable groups today.

Continuing research of the norm of group in capable groups, X.~Guo and X.~Zhang
\cite{Guo:1} in 2012 established necessary and sufficient conditions for the
coinsiding of the norm of the group with its centre, and also investigated the
properties of the norm $N(G)$ in the class of nilpotent groups with cyclic
derived subgroup.

In 2005 N.~Gavioli, L.~Legarreta, S.~Sica, M.~Tota \cite{Gav:1} considered the relations between the centre $Z(G)$, the norm $N(G)$ and the second hypercentre $Z_{2}(G)$ depending on the number $v(G)$ of conjugacy classes of non-normal subgroups and the number $w(G)$ of conjugacy classes of subgroups, which are normalizers of some subgroups, in finite $p$-groups ($p\ne 2$) of nilpotency class $c$.

In 2008 F.~Russo \cite{Rus:1} studied the relations between the centre $Z(G)$, the norm $N(G)$, the quazicenter $Q(G)$ and the hyperquazicenter $Q^{*}(G)$ of quazicentral-by-finite groups. Recall that the quazicenter $Q(G)$ of a group $G$ is the subgroup generated by all elements $x$ of a group $G$ such that the subgroup $\left\langle x\right\rangle $ is permutable in a group $G$ (with other subgroups). Accordingly, the hyperquazicenter $Q^{*}(G)$ of a group $G$ is the largest term of the chain of normal subgroups
$$E=Q_{0}(G)\le Q_{1}(G)=Q(G)\le ...\le Q_{\alpha } \le
Q_{\alpha +1} \le ...,$$
where $\alpha $ is an ordinal and $Q_{\alpha +1}
(G)/Q_{\alpha } (G)=Q(G/Q_{\alpha}(G))$.

\begin{proposition}\label{p:1.5}(\cite{Rus:1}, proposition 3.2) Let $G$ be a quasicentral-by-finite group, $Q(G)$ be the quasicenter of $G$, $N(G)$ be the norm of $G$, then

\textit{1)} if $Q(G)$ contains only elements of prime or infinite order and
$Q(G)=N'$, where N is the subgroup generated by the quasicentral elements of
infinite order, then $G$ is finite;

\textit{2)} if there is an element $x\in N(G)$ such that the index
$\left|Q(G):\left\langle x\right\rangle _{G} \right|$ is finite, then $G$ is
central-by-finite;

\textit{3)} $G$ is central-by-finite if and only if the index
$\left|Q(G):N(G)\right|$ is finite.

\end{proposition}

The relations between the norm $N(G)$ and the center $Z(G)$ in the class of
finite groups have also been studied by I.~V.~Lemeshev in \cite{Lemesh:1}. His
results supplement Baer's results for finite groups.

The study of finite groups, in which Baer norm has a certain index, is very
effective. In particular, in \cite{Wang:1} J.~Wang and X.~Guo studied finite
$p$-groups, in which the norm has a prime index, in \cite{Wang:2} they studied
finite groups, in which the norm is a subgroup of index $p$ or $pq$, where $p$
and $q$ are different prime numbers. J. Smith \cite{Smith:1} studied groups in
which each subgroup of the norm is normal in the group.

Subgroups of an arbitrary group can be considered as elements of some subgroups
lattice $L(G)$ relative to the operations of union and intersection, ordered by
inclusion. In this sense, the norm $N(G)$ of a group can be defined as follows
\cite{Rus:2}:

$$N(G)=\bigcap\limits _{X\in L(G)}N_{G}(X).$$

In this context, in \cite{Rus:2} the relation between the non-cyclicness of the
norm $N(G)$ on the one hand and the subgroup lattice $L(G)$ of the group $G$
and generalized degree of commutativity of the group $G$ on the other hand is
investigated.

A question naturally arises why this characteristic subgroup, in contrast to
the center and the derived subgroup, did not received adequate attention in the
early development of group theory in view of the simple definition of the norm
and its usefullness in the study of groups. G. Miller \cite{Mil:1} explains
that at that time other problems were in algebra and the main focus of group
theory has been directed to the study of solutions of algebraic equations (in
this theory simple groups play a fundamental role, while the norm of a simple
group of composite order is trivial). The norm did not play an important role
also in the study of permutation groups of low degrees, which were used in the
theory of algebraic equations at that time. The smallest degree of permutation
group, which has norm of prime index, is equal to 8, moreover only one of 200
groups of this order has norm of prime index. And perhaps R.Baer drew attention
to this characteristic subgroup only in 1934 for these reasons.

Considering the intersection of normalizers of subgroups of the group, we can
get subgroups associated with Baer norm. These are the intersection of
normalizers of all subgroups contained in the given subgroup
\cite{Sub:1, Sub:2, Scot:1} or conversely the intersection of the normalizers which
contain the given subgroup \cite{Fal:1}. In particular, the concept of
invariator $I_{G} (A)$ of the subgroup $A$ in the group $G$, which was
introduced by I.~Ya.~Subbotin, is closest to the concept of the norm $N(G)$ of
the group $G$.

\textit{Invariator} $I_{G}(A)$ of subgroup $A$ in the group $G$ \cite{Sub:1}
(\textit{quazicenralizer} \cite{Scot:1}) is the intersection of normalizers of
all subgroups of the group $A$ in $G$. This subgroup can also be called the
norm of subgroup $A$ in the group $G$ \cite{Kir:1}. In the case when the
subgroup $A$ coincides with the whole group $G$ the invariator $I_{G}(G)$ is
exactly the norm $N(G)$ of the group $G$.

In 2001 M. De Falco, F. de Giovanni, C. Musella \cite{Fal:1} introduced the
concept of $H$-norm of the group $G$ for some subgroup $H$ of the group $G$.
$H$-norm of a group $G$ is called a subgroup $\ker (G:H)$ consisting of all
elements which normalize every subgroup of $X$ in $G$ containing $H$.
Obviously, if $H\le K\le G$, then $H\le \ker (G:H)\le N_{G} (H)$, $\ker
(G:H)\le \ker (G:K)$. Let's note that the $E$-norm, where $E$ is the identity
subgroup of a group $G$, coincides with the norm $N(G)$ of the group $G$.

It is clear that the norm $N(G)$ can be defined as the subgroup of a group $G$
consisting of all elements of this group, which normalize every subgroup in
$G$. Replacing the condition of normality to pronormality we get some analogue
of the norm of a group for pronormality. It is called \textit{pronorm} $P(G)$.

Recall that an element $x$ of the group $G$ \textit{pronormalizes subgroup} $H$
of a group $G$, if subgroups $H$ and $H^{x}$ are conjugate in $\left\langle
H,H^{x} \right\rangle$. Accordingly, the \textit{pronorm $P(G)$} of a group $G$
is the set of all elements of a group $G$ which pronormalize every subgroup of
a group. For the first time the concept of pronorm $P(G)$ group was introduced
by F.~de~Giovanni, S.~Vincenzi \cite{Giov:1} in 2000.

In contrast to the norm of a group the pronorm is not always a subgroup of a
group. In \cite{Roman:1} some classes of groups, in which the set of all elements
of a group $G$ that pronormalize every subgroup of a group, forms a subgroup,
were investigated.
\begin{proposition}\label{p:1.6} (\cite{Giov:1})If $G$ is polycyclic group, then its pronorm $P(G)$ is a subgroup.
\end{proposition}
In this paper a similar statement for the class of locally soluble groups was
proved.

Subgroups generated by normalizers of given subgroups are considered in some
researches, concerning groups with restrictions on normalizers of given systems
of subgroups. In this context, let's note the research of J. Smith
\cite{Smit:2}, who studied the subgroup $R=R(G)$ generated by all proper
normalizers, and called it \textit{conorm of a group}. If the group $G$ has not
proper normalizers, then the group $G$ is Dedekind and $R(G)=E$.

In 1990 H.~Bell, F.~Guzman, L.-Ch.~Kappe \cite{Bell:1} studied so-called
Baer-kernel, which is ring analogue of the norm of the group. Baer-kernel of
the ring $K$ is defined as the set

\centerline{$B(K)=\{a\in K|\forall y\in K,\exists r,s\in N(ay=y^{r}a\wedge
ya=ay^{s})\}.$}

In 2010 year M.~R.~Dixon, L.~A.~Kurdachenko, D.~Otal used the so-called norm of
subspace in linear groups in the research of linear groups with finite
dimensional orbits \cite{Dix:1}.

Let $A$ be a vector space over a field $F$, $GL(F,A)$ be a group of all
automorphisms of a space $A$, $G$ be a subgroup of a group $GL(F,A)$, $B$ be a
subspace of a space $A$. The norm of the subspace $B$ in the group $G$ is the
intersection of normalizers of all $F$-subspases in $B$:

$$Norm_{G} (B)=\bigcap\limits _{b\in B}N_{G}(bF).$$

It is known when the group $G$ coincides with the norm $Norm_{G}(B)$, then the
group $G$ is isomorphic to a subgroup of the multiplicative group $U(F)$. If
the group $G$ has finite dimensional orbits over $A$, then $A$ contains a
$FG$-submodule $D$ of finite dimension $\dim _{F}(D)$. If $K=C_{G}(D)$, then
$K\le Norm_{G} (A/D)$. When $G$-orbits of every subspace from $A$ are finite,
then \textit{A} contains a $FG$-submodule $B$ such that $\dim _{F}(A/B)$ and
$\left|G:Norm_{G} (B)\right|$ are also finite.

Therefore, the research devoted to the study of the norms of the group and
related subgroups is very important and interesting direction in the group
theory. At the same time, there are many questions regarding the structural
characteristics of the group depending on the structure of its norms,
conditions of coinsiding of the norm of the group and its center, etc.

\section{Generalized norms of some systems of maximal and subnormal subgroups}

As noted above, the norm $N(G)$ is the $\Sigma $-norm of the group, in which
the system $\Sigma $ is a system of all subgroups of this group. Narrowing the
system of all subgroups, for example, to the system of all Abelian or all
maximal subgroups of a group, we will receive new $\Sigma $-norms, which can be
considered as generalizations of the norm $N(G).$

The first generalizations of this kind appeared in the 50-th of XX century. In
particular, in 1953 R.~Baer \cite{Baer:11} considered the intersection $H(G)$
of normalizers of all Sylow subgroups of a group $G$ and called this intersection as
\textit{hypercenter of a group} $G$. It is clear that hypercenter $H(G)$ is the
$\Sigma $-norm, where the system $\Sigma $ consists of all Sylow subgroups of
the group. R.~Baer proved that $H(G)$ coincides with the intersection of all
maximal nilpotent subgroups, and the quotient group $G/H(G)$ is a group with
trivial center. Moreover, it was found that the normal subgroup belongs to
hypercenter if and only if its elements of order $p^{n} $ generate cyclic
subgroups of index $p^{n} $.

In 1968 B.~Huppert \cite{Hup:1} generalized the concept of hypercenter
introducing the concept of \textit{$\Im$-hypercenter}. Let $\Im$ be a
class of finite groups which can be represented as direct products of their
Hall $\pi$-subgroups with respect to some partition of non-empty set $\pi$ of
all primes. This class is a local formation. The chief factor $H/K$ of a group
$G$ is called $\Im$-central \cite{Shem:1}, if $H/K\lambda (G/C_{G} (H/K))\in
\Im$. The product of all normal subgroups of $G$ which $G$-chief factors are
$\Im$-central in $G$ is called \textit{$\Im$-hypercenter $Z_{\Im}(G)$}
of a group $G$ \cite{Doerk:1}. In 2013 V.~I.~Murashka \cite{Mur:1} studied the
properties of $\Im$-hypercenter and got some Baer's results on the norm of
the group as corollaries in some cases.

One of the mentioned generalizations of the norm of the group is so-called
$A$-norm $N_{A}(G)$ of the group $G$. It is the intersection of normalizers of
all maximal Abelian subgroups. This norm was introduced by W.~Kappe
\cite{Kap:1} in 1961. As it turned out (see \cite{Kap:1}) in finite group
$A$-norm is a subgroup, each element of which is permutable with its conjugate
(such groups were studied, in particular, by F.~Levi \cite{Lev:1}). In
addition, it was found that the $A$-norm is close to a subgroup of right Engel
elements of length 2, that allowed to use it in the study of Engel groups.

Recall (see e.g. \cite{Kur:1}) that the element $x\in G$ is called the
\textit{right Engel element of length 2}, if for arbitrary element $g\in G$
there is a relation $\left[\left[x,\; g\right],\; g\right]=1$.

Let $R(G)$ denote the subgroup of a group $G$ generated by all
right Engel elements of length 2 of a group $G$. Next propositions take
place.

\begin{proposition}\label{p:2.1} (\cite{Kap:1}) $A$-norm N$_{A}(G)$ of a group $G$
contains the second hypercnter of a group $G$ and is contained in the subgroup $R(G)$. Moreover the quotient group $R(G)/ N_{A}(G)$ is elementary Abelian
group of exponent not exeeding 2.\end{proposition}

\begin{proposition}\label{p:2.2} (\cite{Kap:1}) For an element $x\in G$ which
order is not divisible by 2, the following statements are equivalent:

\textit{1)} $x\in N_{A}(G)$;

\textit{2)} $x$ is right Engel element of length 2 in $G$;

\textit{3)} if $\left\langle x\right\rangle \triangleleft G$ and $U$ is the group
of automorphisms induced on $\left\langle x\right\rangle$ by $G$, then $x$
belongs to $A$-norm of the group $\left\langle x\right\rangle U$;

\textit{4)} for arbitrary elements $g,h\in G$ the equality $\left[\left[x,
g\right], h\right]=\left[\left[x, h\right],g\right]^{-1}$ takes place.\end{proposition}

The following proposition on $A$-norm is a generalization of Wos' \cite{Wos:1}
and Schenkman's results \cite{Schen:1} on the norm $N(G)$ of the group.

\begin{proposition}\label{p:2.3} (\cite{Kap:1}) Group $G$ induces on the subgroup
$N_{A}(G)$ a nilpotent group of automorphisms. Its class of nilpotency does
not exceed 2.\end{proposition}

Further W.~Kappe [\ref{Kap:2}--\ref{Kap:4}] have generalized the concept of the
$A$-norm of the group and introduced so-called $E$-norm, which was defined as
the intersection of normalizers of all maximal subgroups of the group with the
given theoretical group property $E$. Clearly, $E$-norm $N_{E}(G)$ contains the
norm $N(G)$. The intersection of arbitrary subgroup of a group $G$ and the
$E$-norm of the group is contained in the $E$-norm of this subgroup. Besides
$N_{E}(N_{E}(G))= N_{E}(G)$.

A subgroup $\Delta (G)$ is related with the concept of the $E $-norm. It was
studied by W.~Gashutz \cite{Gash:1} and was defined as the intersection of
normalizers of all maximal subgroups of the group. It is clear that Gashutz
subgroup $\Delta (G)$ can be considered as $\Sigma$-norm of a group for the
system $\Sigma$ that consists of non-normal in $G$ maximal subgroups. In
\cite{Gash:1} it was found that $\Delta (G)$ is nilpotent and $\Delta(G)/\Phi
(G)=Z(G/\Phi (G))$, where $\Phi (G)$ is Frattini subgroup.

In 1958 H.~Wielandt \cite{Wiel:1} studied the properties of normalizers of
subnormal subgroups and introduced the subgroup $W(G)$. It is the intersection
of normalizers of all subnormal subgroups of a group. It is clear that
\textit{Wielandt subgroup $W(G)$} is the \textit{norm of subnormal subgroups}
of a group.

It is obvious that subnormal norm coincides with the norm $N(G)$ in
nilpotent group. In addition, the condition $G=W(G)$ is equivalent to the fact
that all subnormal subgroups of a group are normal. By theorem 13.3.7 \cite{Rob:1}
Wielandt subgroup $W(G)$ contains every simple non-Abelian subnormal
subgroup of $G$ and every minimal normal subgroup of $G$ which
satisfies the minimal condition for subnormal subgroups. Therefore, the
subgroup $W(G)$ is not trivial in finite group $G$ \cite{Wiel:1}.

D.~Robinson \cite{Rob:2} and J.~Roseblade \cite{Rose:1} independently from each other got
similar result for some classes of infinite groups.

\begin{proposition}\label{p:2.4} (\cite{Rob:2, Rose:1}). If a group $G$ satises the minimal condition for subnormal subgroups, then the quotient group $G/W(G)$ is finite.
\end{proposition}

These results were summarized by J.~Cossey \cite{Coss:1} for polycyclic groups
and he found that these groups have a finite quotient group $G/C_{G}(W(G)).$

Wieland subgroup and its generalizations has been studied intensively by O.~Kegel \cite{Keg:1}, J.~Cossey, R.~Bryce \cite{Bryce:6, Bryce:7, Bryce:8}, R.~Brandl, F.~Giovanni, S.~Franciosi \cite{Bran:1}. A.~Camina \cite{Cam:1}, C.~Casolo \cite{Casol:1, Casol:2}, E.~Ormerod \cite{Orm:1}, C.~Wetherell \cite{Weth:1, Weth:2}, X.~Zhang and X.~Guo \cite{Zhan:1, Guo:2}.

In \cite{Bryce:6} it was proved that the subnormal norm $W(G)$ is contained in the $FC$-centre in finitely generated soluble-by-finite group of finite rank. Furthermore, if the norm $W(G)$ coincides with the whole group, then all subnormal subgroups are normal in this group, that is, the normality is transitive relation. Groups with such property were studied by D.~Robinson in \cite{Rob:3} and were called $T$-groups. If $G$ is a finite soluble $T$-group and $G/L$ is the unique maximal nilpotent quotient group of group $G$, then the quotient group $G/L$ is Abelian or Hamiltonian and $L$ is Abelian.

In 1989 J.~Cossey, R.~Bryce \cite{Bryce:6} introduced \textit{local Wielandt subgroup} $W^{p}(G)$ that is the intersection of normalizers of all $p'$-perfect subnormal subgroup of a group $G$. Recall that the $p'$-perfect group is a group that has no non-trivial quotient groups of order coprime with $p$.

In 1992 C.~Casolo \cite{Casol:2} studied a special subgroup of a group $W(G)$,
which was called \textit{strong Wielandt subgroup} $\overline{W}(G)$, and
defined as the intersection of the centralizers of nilpotent subnormal quotient
groups of the group $G$: $$\overline{W}(G)=\left\{ g\in G|\left[S,g\right]\le
S^{R}\ \ \text{for all} \ \ S<<G \right\},$$ where $S^{R} $ is nilpotent
residual of subgroup $S$ or the smallest normal subgroup $N$ of $S$ such that
the quotient group $S/N$ is nilpotent. C.~Casolo proved that strong Wielandt
subgroup $\overline{W}(G)$ is non-trivial in finite group. Note that this
subgroup was also studied by C.~Wetherell \cite{Weth:1, Weth:2}.

In 1990 R.~Bryce \cite{Bryce:8} introduced one more generalization of Wielandt
subgroup, so-called $m$-\textit{Wielandt subgroup} $U_{m}(G)$ of a group $G$
that is the intersection of normalizers of all subnormal subgroups of a group
$G$ with defect at most $m$ for integer $m\ge 1$. He investigated polynilpotent
lattice of finite soluble groups in terms of Wielandt $m$-length. The concept
of $m$-series of Wielandt group is widely used. It is defined as follows: for
each natural $m\ge 1, U_{m,0}(G)=E$; if $i\ge 1$, then $U_{m,i} (G)$ is
determined from the condition

$$U_{m,i} (G)/U_{m,i-1} (G)=U_{m} (G/U_{m,i-1} (G)).$$

If $U_{m,n} (G)=G$ for some integer $n$, then such minimal number $n$ is called Wielandt $m$-length. R.~Bryce proved that there are limits of commutator length and Fitting length of finite soluble groups in terms of Wielandt $m$-length ($m\ge 2$), and identified the best such restriction. Properties of \textit{Wielandt $m$-subgroup} $U_{m} (G)$ have also been studied by C.~Franchi \cite{Fran:1, Fran:2}.

In 1995 J.~Biedleman, M.~Dixon, D.~Robinson \cite{Biedl:1,Biedl:2} considered
one more $\Sigma$-norm of a group -- \textit{generalized Wielandt subgroup
$IW(G)$} which is the intersection of normalizers of all infinite subnormal
subgroups of a group. It is clear that $IW(G)$ is a characteristic subgroup and
contains a subnormal norm $W(G)$. If $G=IW(G)$, then all infinite subnormal
subgroups are normal in the group. Such groups have been studied by
F.~Giovanni, S.~Franciosi \cite{Giov:2} and was called \textit{$IT$-groups}. In
\cite{Biedl:1} the structure of the group $G$ with the property $IW(G)\ne W(G)$
and the structure of the quotient group $IW(G)/W(G)$ were investigated.

In \cite{Mari:1} F.~Mari, F.~Giovanni introduced a new $\Sigma $-norm, in which system $\Sigma $ consists of all nonsubnormal subgroups of a group. This \textit{norm of nonsubnormal subgroups} was denoted by $W^{*}(G)$. It is clear that if $W^{*}(G)=G$, then all subgroups are subnormal in a group. Moreover, if a group $G$ is a group with a finite number of normalizers of subnormal subgroups, then the quotient group $G/W^{*}(G)$ is finite \cite{Mari:1}.

Let's also mention the research \cite{Amin:1}, in which so-called \textit{generalized $N$-Wielandt subgroup} $W_{N}(G)$
was introduced. It consists of all elements of the group $G$, which normalize all subnormal subgroups of $N$. It is a normal subgroup and, in general, may differ from $N$.

It is clear that $W(G)\subseteq W_{N}(G)$, in particular, $W(G)=W_{N}(G)$, if $N=G$, or $N=W(G)$, or $N$ is the unique maximal normal subgroup. If $G$ is a $T$-group and $N$ is a normal subgroup of $G$, then $W_{N}(G)=G$. The following example proves that the converse is not true.

\begin{example} \cite{Amin:1} Let $G=D_{8} =\left\langle x,y\right\rangle $, $x^{8} =y^{2}=(xy)^{2} =1$, $N_{1}=\left\langle x^{2}\right\rangle$, $N_{2}=\left\langle x\right\rangle$, then $W_{N_{1}}(G)=W_{N_{2}}(G)=G$, but $G$ is not a $T$-group. \end{example}

\section{Generalized norms of characteristic subgroups of a \\group}

Nowadays algebraists turn attention to a generalization of the norm when the
system $\Sigma$ is selected as a system of some characteristic subgroups. In this
context Sh.~Lia and Zh.~Shen \cite{Lia:1, Shen:1} considered the $\Sigma$-norm
$D(G)$ of a finite group, where the system $\Sigma $ is chosen as a system of
derived subgroups of all subgroups of the group. The authors proved that in the
case when $D(G)$ contains all the elements of prime order, the group $G$ is
solvable of Fitting length at most 3. In the case when $G=D(G)$, derived
subgroup $G'$ is nilpotent and $G''$ has nilpotency class at most 2.

Recently a number of researches concern the norms of different systems
of residuals. In particular, Zh.~Shen, W.~Shi and G.~Qian \cite{Shen:2} studied
the \textit{norm $S(G)$ of nilpotent residuals of all subgroups of prime order}. It was
proved that if all elements of prime order of a finite group $G$ are contained
in the norm $S(G)$, then the group $G$ is solvable. L.~Gong and X.~Guo
\cite{Gong:1} studied the norm of nilpotent residuals of all subgroups of a
finite group. N.~Su and Ya.~Wang \cite{Su:1} considered the norm
$D^{\mathfrak{F}}(G)$ of \textit{$\mathfrak{F}$-residual} $G^{\mathfrak{F}}$ of
all subgroups of the group $G$ and the norm
$D_{p}^{\mathfrak{F}}(G)\mathfrak{H}^{\mathfrak{F}}O_{p'}(G)$ of all subgroups
$H$ of a finite group $G$, where $\mathfrak{F}$ is the formation. Recall that
$\mathfrak{F}$-residual $G^{\mathfrak{F}}$ of a group $G$ is the smallest
normal subgroup $N$ of $G$ such that $G/N\in \mathfrak{F}$.

X.~Chen and W.~Guo \cite{Chen:1} introduced the $\mathfrak{h}\mathfrak{F}$-norm
$N_{\mathfrak{h}\mathfrak{F}}(G)$ of group $G$. It is the intersection of
normalizers of products of $\mathfrak{F}$-residuals of all subgroups of a group
$G$ and $\mathfrak{h}$-radical of group $G$

$$N_{\mathfrak{h,F}}(G)=\bigcap\limits_{H\le
G}N_{G}(H^{\mathfrak{F}}G_{\mathfrak{h}}),$$
\noindent where $\mathfrak{h}$ is
Fitting class, $\mathfrak{F}$ is formation. Recall that
$\mathfrak{h}$-radical $G_{\mathfrak{h}}$ of a group $G$ is maximal normal
$\mathfrak{h}$-subgroup of a group $G$.

If $\mathfrak{h}=1$, then the subgroup $N_{1,\mathfrak{F}}(G)$ is called
$\mathfrak{F}$-norm $N_{\mathfrak{F}}(G)$ of a group $G$ and defined as
$$N_{\mathfrak{F}}(G)=\bigcap\limits_{H\le G}N_{G}(H^{\mathfrak{F}}).$$ If
$\mathfrak{h}=\mathfrak{G}_{\pi}$, where $\mathfrak{G}_{\pi}$ is the class of
finite $\pi$-solvable groups, then the subgroup $N_{\mathfrak{G}_{\pi
},\mathfrak{F}}(G)$ is called $\pi\mathfrak{F}$-norm $N_{\pi\mathfrak{F}}(G)$
of a group $G$ and defined as $$N_{\pi\mathfrak{F}}(G)=\bigcap\limits_{H\le
G}N_{G}(H^{\mathfrak{F}}O_{\pi}(G)).$$

X.~Chen and W.~Guo investigated the
properties of $\mathfrak{h}\mathfrak{F}$-norm, in particular,
$\pi\mathfrak{F}$-norm of a finite group $G$ and the relations between
$\pi'\mathfrak{F}$-norm and $\pi\mathfrak{F}$-hypercentre of a group $G$.

In 2014 A.~Ballester-Bolinches, J.~Cossey, L.~Zhang \cite{Ball:1} proposed to
generalize the structure of $\Sigma$-norms which appeared recently. The
authors defined the \textit{$C$-norm} $kC(G)$ of a finite group $G$ as the
intersection of the normalizers of all subgroups of the group $G$ which
\textit{do not belong to the class $C$}

$$k_{C}(G)=\bigcap\limits_{H\notin C}N_{G}(H)$$ provided that $k_{C}(G)=G$,
\noindent if $G\in C$. With this approach Baer norm $N(G)$ can be considered
as the norm $k_{C}(G)$, where \textit{C} is the class of groups of order 1.
Groups with $k_{C}(G)=G$ are called $C$-Dedekind. In \cite{Ball:1} the structure
of non-nilpotent $C$-Dedekind groups for the class of nilpotent groups is
described. It is also shown that the groups, which $C$-norm is not
hypercentral, have a very restricted structure. The authors gave the
classification of nilpotent classes closed under subgroups, quotient groups and
direct products of groups of coprime orders, and showed that the known
classifications can be deduced from this one.

\begin{proposition} (\cite{Ball:1}) If $k_{C}(G)$ contains non-central chief
factor of $G$, then $k_{C}(G)$ contains exactly one non-central chief factor
(in any chief series through $k_{C}(G)$ of a group $G$) and if $p$ is a prime
divisor of the order of this chief factor, then Hall $p'$-subgroup of $G$ is
$C$-group and $G$ has nilpotency class at most 3.
\end{proposition}

Note also R.~Laue's research \cite{Lau:1}. He considered a subgroup closed to
the $\Sigma $-norm

\centerline{$A(\Sigma)=\bigcap\limits_{X\in \Sigma }N_{Aut(G)}(X),$} \noindent which
consists of automorphisms that normalize every $\Sigma $-subgroup of a group
$G$.

\section {Generalized norms of different systems of Abelian and non-cyclic subgroups}

The narrowing of a system $\Sigma $ of all subgroups of the group \textit{G} to
the system of all Abelian and all cyclic subgroups does not lead to extension
of the concept of the norm $N(G)$. However, when the system $\Sigma$ is the system
of all non-cyclic subgroups (provided that such subgroups exist in the group),
then such $\Sigma$-norm (let's call it the \textit{norm of non-cyclic
subgroups}) differs from the norm $N(G)$ in general case. The opportunity to
study the norm of non-cyclic subgroups was given due to F.~M.~Lyman's research
\cite{Lym:1,Lym:2,Lym:3}. He received a description of some classes of non-Abelian
groups in which all non-cyclic subgroups are normal. These groups were called
$\overline{H}$-\textit{groups} ($\overline{H_{p}}$-\textit{groups} in the case
of $p$-groups).

The concept of the \textit{non-cyclic norm} $N_{G}$ of a group as the intersection
of the normalizers of all non-cyclic subgroups of the group was introduced by
F.~M.~Lyman in 1997 \cite{Lym:4}, where he studied infinite groups, in which
non-cyclic norm is locally-graded and has a finite index.

\begin {proposition} (\cite{Lym:4}) In the group $G$ non-cyclic norm
is locally-graded and has finite index if and only if the group $G$ is
central-by-finite.
\end{proposition}

In addition, it was proved that for the condition $1<\left|G/N_{G}
\right|<\infty $ in the class of infinite locally finite groups the non-cyclic
norm \textit{N${}_{G}$} is Dedekind, and in the class of non-periodic locally
soluble-by-finite groups it is Abelian \cite{Lym:4}.

The study of the non-cyclic norm was continued by F.~M.~Lyman and
T.~D.~Lukashova in \cite{Luk:1,Luk:2,Luk:3,Luk:4,Lym:5}, where the authors characterized the
structure of wide classes of groups, which non-cyclic norm is non-Dedekind.
Since O.~Yu.~Olshansky infinite groups \cite{Ols:1}, all subgroups of which
are cyclic and which are the norms of their non-cyclic subgroups, exist,
periodic groups were considered by the authors provided their local finiteness.
In particular, in \cite{Shen:3} it was proved that the class of infinite
locally finite $p$-groups ($p\neq 2$), in which non-cyclic norm $N_{G}$ is
non-Abelian, coincides with the class of non-Abelian $p$-groups, all non-cyclic
subgroups of which are normal. At the same time, there are infinite locally
finite 2-groups which have proper non-Dedekind norm of non-cyclic subgroups.
The structure of locally finite $p$-groups ($p$ is prime), which non-cyclic
norm is non-Dedekind, is described in \cite{Luk:1,Luk:2,Luk:3}.

\begin{proposition} (\cite{Luk:1}) Locally finite $p$-groups ($p\neq 2$),
which have non-Abelian non-cyclic norm $N_{G}$, are groups of the following
types:

\textit{1)} $G$ is $\overline{H}_{p}$-group, $N_{G}=G$;

\textit{2)} $G=\left(\left\langle x\right\rangle \times \left\langle
b\right\rangle \right)\lambda \left\langle c\right\rangle,\left|x\right|=p^{n},
n>1,\left|b\right|=\left|c\right|=p, \left[b,c\right]=x^{p^{n-1}},
\left[x,c\right]=x^{p^{n-1} \alpha}b^{\beta}$, $(\beta,p)=1;
N_{G}=\left(\left\langle x^{p} \right\rangle \times \left\langle b\right\rangle
\right)\lambda \left\langle c\right\rangle $;

\textit{3)} $G=\left\langle x\right\rangle \left\langle
b\right\rangle,\left|x\right|=p^{k},\left|b\right|=p^{m},m>1,k\ge
m+r,Z(G)=\left\langle x^{p^{r+1} } \right\rangle \times \left\langle
b^{p^{r+1}}\right\rangle,1\le r\le m-1$,
$\left[x,b\right]=x^{p^{k-r-1}s}b^{p^{m-1}t}, (s,p)=1, N_{G}=\left\langle
x^{p^{r}}\right\rangle \lambda \left\langle b\right\rangle$.

\end{proposition}

\begin{proposition}(\cite {Luk:2,Luk:3} Locally finite 2-groups $G$
with non-Dedekind non-cyclic norm $N_{G}$ are groups of the following types:

\textit{1)} $G$ is non-Hamiltonian $\bar{H}_{2}$-group, $G= N_{G}$;

\textit{2)} $G=\left(A\times \left\langle b\right\rangle \right)\lambda
\left\langle c\right\rangle \lambda \left\langle d\right\rangle$, $A$ is
quasicyclic 2-group, $\left[A,\left\langle c\right\rangle \right]=1,$
$\left|b\right|=\left|c\right|=\left|d\right|=2$, $d^{-1}ad=a^{-1}$ for
arbitrary element $a\in A$,
$\left[b,c\right]=\left[d,b\right]=$$\left[d,c\right]=a_{1}$, $a_{1}\in A,
\left|a_{1}\right|=2$; $N_{G}=\left(\left\langle a\right\rangle \times
\left\langle b\right\rangle \right)\lambda \left\langle c\right\rangle $, where
$a \in A,\left|a \right|=4$;

\textit{3)} $G=(A \times H)\left\langle d\right\rangle\, A$ is quasicyclic
2-group, $d^{2}=a_{1}\in A, \mid{a_{1}}\mid =2, d^{-1}ad=a^{-1}$ for arbitrary
element $a \in A$, $H=\left\langle h_{1},h_{2}\right\rangle,\left|h_{1}
\right|=\left|h_{2} \right|=4, h_{1}^{2}=h_{2}^{2}=\left[h_{1},h_{2}\right]$,
$\left[d,h_{1} \right]=a_{1},\left[d,h_{2}\right]=1$; $N_{G}=\left\langle h_{2}
\right\rangle \lambda \left\langle h_{1}a\right\rangle,\left|a\right|=4, a \in
A$, $\mid a\mid=4$;

\textit{4)} $G=(<x>\times<b>)\lambda<c>\lambda<d>,
|x|=2^{n}$,$n>2,|b|=|c|=|d|=2,[x,c]=1, d^{-1}xd=x^{-1}$,
$\left[b,c\right]=\left[d,b\right]=\left[d,c\right]=x^{2^{n-1}}$;
$N_{G}=(<x^{2^{n-2}}>\times <b>)\lambda \left\langle c\right\rangle$;

\textit{5)} $G=(<x>\lambda<b>)\lambda<c>, |x|=2^{n}, n>3,
|b|=|c|=2,[x,c]=x^{\pm 2^{n-2}}b, [x,b]=x^{2^{n-1}}$;
$N_{G}=(<x^{2}>\times<b>)\lambda<c>$;

\textit{6)} $G=<x>\lambda H, |x|=2^{n}$, $n>2, H=<h_{1},h_{2}>,|h_{1}|=|
h_{1}|=4$, $h_{1}^{2}=h_{2}^{2}=[h_{1},h_{2}], [<x>,H]=<x^{2^{n-1}}>;
N_{G}=<x^{2}>\times H$;

\textit{7)} $G=(<x>\times H)<y>, |x|=2^{n}, n\geq2, H=<h_{1},h_{2}>, |h_{1}|=
|h_{2}| =4, h_{1}^{2}=h_{2}^{2}=[h_{1},h_{2}], y^{2}=x^{2^{n-1}},
[y,h_{2}]=1,[y,h_{1}]=y^{2}, y^{-1}xy=x^{-1}$;
$N_{G}=<h_{2}>\lambda<h_{1}x^{2^{n-2}}>$;

\textit{8)} $G=\left\langle x\right\rangle \left\langle b\right\rangle
,\left|x\right|=2^{k},\left|b\right|=2^{m}, m\ge 1$; if $m=1$, then $k=3$,
$[x,b]=x^{2}$ and $N_{G}=<x^2>\lambda<b>;$ if $m>1$, then $k \geq m+r, 1\le
r\le m-1, Z(G)=<x^{2^{r+1}}>\times <b^{2^{r+1}}>, \left[x,b\right]=x^{2^{k-r-1}
s}b^{2^{m-1}t}$, $0<s<2, 0 \leq t<2,$ ($k>3$ and $t=0$ if $m=2$);
$N_{G}=\left\langle x^{2^{r}} \right\rangle \lambda \left\langle b\right\rangle
$.
\end{proposition}

Non-primary locally finite groups with non-Dedekind non-cyclic norm were
studied in \cite{Luk:2, Luk:5}. It was found that infinite locally finite
non-primary groups with such restrictions on the norm $N_{G}$ is locally
nilpotent.

\begin{proposition} (\cite{Luk:2}) Infinite locally finite
non-primary groups with non-Dedekind non-cyclic norm $N_{G}$, are locally
nilpotent and are groups of the following types:

\textit{1)} $G$ is infinite non-primary non-Hamiltonian $\bar H$--group,
$G=N_{G}$;

\textit{2)}$G=G_{2}\times \left\langle y\right\rangle, G_{2}$ is a group of one
type 2) or 3) of proposition 4.3, ($|y|,2)=1; N_{G}=N_{G_{2}}\times
\left\langle y\right\rangle$.
\end{proposition}

\noindent Thus, a locally finite group, which non-cyclic norm $N_{G}$ is non-nilpotent, is finite.

Continuing the study of locally finite groups with non-Dedekind non-cyclic
norm, in \cite{Luk:4} it was proved that finite nilpotent groups with such
restrictions are groups of the type

\centerline {$G=G_{p} \times \left\langle y\right\rangle,$} \noindent where
$G_{p} $ is Sylow $p$-subgroup of a group $G$ and a finite group with a
non-trivial norm $N_{G_{p}}$, ($|y|,p)=1$. In addition, if the non-cyclic
norm $N_{G}$ is non-nilpotent in the class of locally finite groups, then all
non-cyclic subgroups in a group are normal.

Non-periodic locally soluble-by-finite groups with non-Dedekind non-cyclic norm
are considered in \cite{Lym:5}.

\begin {proposition} (\cite{Lym:5}) Any non-periodic locally
soluble-by-finite group $G$ that has non-Dedekind non-cyclic norm $N_{G}$ is
$\overline{H}$-group and $G=N_{G}$.
\end{proposition}

Note that locally finite or non-periodic locally soluble-by-finite groups with
non-Dedekind norm of non-cyclic subgroups are soluble and their degree of
solvability does not exceed 3 by the results of \cite{Luk:1,Luk:2,Luk:3,Luk:4,Lym:5}.

Zh.~Shen, W.~Shi, J.~Zhang \cite{Shen:3,Shen:4} studied the properties of the
norm $N_{G}$ of non-cyclic subgroups in the class of finite groups and its
impact on the group. The authors proved that the norm of non-cyclic subgroups
of a finite group is soluble. Note that this proposition is a direct corollary
from the description of finite $\overline{H}$-groups (see \cite{Lym:1,Lym:2,
Lym:3}). It was also proved that a finite group is soluble if all its elements
of prime order are contained in norm $N_{G}$ of non-cyclic subgroups. In
addition, it was found that the derived subgroup is nilpotent if all elements
of prime order or of order 4 of a group are contained in $N_{G}$ \cite{Shen:3}.

\begin{proposition}(\cite{Shen:3}) Finite group has nilpotent derived
subgroup if and only if derived subgroup of quotient group through norm $N_{G}$
is also nilpotent.
\end{proposition}

The study of infinite groups with given restrictions on normalizers of
different systems $\Sigma$  of infinite subgroups have been the subject of many
theoretical-group researches for a long time. Therefore, when considering
infinite groups with restrictions on $\Sigma$-norm, it is naturally to choose
one of systems of infinite subgroups as a system $\Sigma$.

In this context, in the study of $\Sigma$-norms of infinite groups F.~M.~Lyman
and T.~D.~Lukashova \cite{Lym:5, Lym:6,Lym:7,Lym:8} considered systems of all
infinite, all infinite Abelian and all infinite cyclic subgroups, provided that
these systems are non-empty. These \textit{$\Sigma$}-norms were denoted as
follows: $N_{G}(\infty)$ is \textit{the norm of infinite subgroups of a group} $G; N_{G}
(A_{\infty})$ is \textit{the norm of infinite Abelian subgroups of a group $G;
N_{G}(C_{\infty })$ is the norm of infinite cyclic subgroups of a group $G$}.

If the group $G$ coincides with one of these $\Sigma$-norms, then all
$\Sigma$-subgroups are normal in it. Infinite non-Abelian groups with the
property $N_{G}(\infty)=G$ and $N_{G}(A_{\infty})=G$ (if such subgroups
exist in them) were studied by S.~M.~Chernikov \cite{Chern:1, Chern:2} and
called $INH$-\textit{groups} and $IH$-\textit{groups} respectively.

Restrictions, which these $\Sigma$-norms satisfied, were non-Dedekindness of
$\Sigma$-norm or finiteness of its index in the group. The following
proposition gives sufficient conditions of Dedekindness of each of these norms.

\begin{proposition} (\cite{Lym:6}) In non-periodic groups the norm
$N_{G}(\infty)$ of infinite subgroups, the norm $N_{G}(A_{\infty})$ of
Abelian infinite subgroups, the norm $N_{G}(C_{\infty})$ of infinite cyclic subgroups are
Dedekind in each of the following cases:

\textit{1)} $G$ is a torsion free group or a mixed group without involution;

\textit{2)} the center of a group $G$ contains elements of infinite order;

\textit{3)} group $G$ is central-by-finite;

\textit{4)} these norms are finite;

\textit{5)} group $G$ contains a subgroup $M$ from the system $\Sigma$ such
that $M\bigcap N_{G}(\Sigma)=E$.
\end{proposition}

The problem of the relations between these norms in non-periodic groups is
quite interesting. The following relation implies from the above definitions

$$Z(G)\subseteq N(G)\subseteq N_{G}(\infty )\subseteq
N_{G}(A_{\infty })\subseteq N_{G}(C_{\infty}).$$
So the natural question is: under what conditions do these norms coincide? The following
proposition gives the answer (in terms of sufficient conditions).

\begin {proposition} (\cite{Lym:6}) In non-periodic group $G$ the equality
takes place

\centerline {$N(G)=N_{G}(\infty )=N_{G}(A_{\infty })=N_{G}(C_{\infty})$}
\noindent provided that at least one of the statements takes place:

\textit{1)} the center of a group $G$ contains elements of infinite order;

\textit{2)} $G$ is torsion free group;

\textit{3)} group $G$ is central-by-finite.

\end{proposition}

Infinite groups with restrictions on the norm $N_{G}\left(\infty \right)$ of
infinite subgroups were studied in \cite{Lym:6}. It turned out that
non-periodic groups, which norm $N_{G}\left(\infty \right)$ has finite index,
are mixed and are finite extensions of their centres.

It was also proved that the norm $N_{G}\left(\infty \right)$ of infinite subgroups of the non-periodic
group is Abelian and coincides with the center of the group, if it contains
elements of infinite order. This result generalizes Baer's theorem
\cite{Baer:9} on the coincidence of the norm $N\left(G\right)$ of the group and
its center in the case of non-periodic norm $N\left(G\right)$. Infinite locally
finite groups, which norm $N_{G}\left(\infty \right)$ is non-Dedekind, are
finite extension of quazicyclic subgroup, which is a divisible part of the norm
$N_{G} \left(\infty \right)$ \cite{Lym:7}.

The structure of non-periodic groups, which norm $N_{G}\left(A_{\infty }
\right)$ of infinite Abelian subgroups is $IH$-group, are characterized by the
following proposition.

\begin{proposition} (\cite{Lym:5}) Non-periodic group $G$ has
non-Abelian norm $N_{G}\left(A_{\infty}\right)$ of infinite Abelian subgroups,
if and only if all elements of infinite order of a group $G$ generate Abelian
normal subgroup $D$ that contains every infinite Abelian subgroup of a group
$G$ and there is an element b of order 2 or 4, such that $b^{-1}db=d^{-1}$ for
an arbitrary element $d\in D$. Moreover $N_{G}\left(A_{\infty}
\right)=D\left\langle b\right\rangle $.

\end{proposition}

A natural generalization of Baer norm for non-periodic groups is the norm
$N_{G}(C_{\infty})$ of infinite cyclic subgroups. The study of this norm and
its impact on properties of the group was started by F.~M.~Lyman and
T.~D.~Lukashova in \cite{Lym:8}. It was proved that the norm
$N_{G}(C_{\infty})$ coincides with the center of the group in torsion free
groups, and any finite over the norm $N_{G}(C_{\infty})$ torsion free group is
Abelian. The following proposition characterizes the properties of the group
that has non-Abelian norm $N_{G}(C_{\infty})$.

\begin{proposition} (\cite{Lym:8}) Non-periodic group $G$ has non-Abelian norm
$N_{G}(C_{\infty})$, if and only if all elements of infinite order of the group
$G$ generate an Abelian normal subgroup $A$ and there is an element $b$ of
order 2 or 4, such that $b^{-1}ab=a^{-1}$ for an arbitrary element $a\in A$.
Moreover $N_{G}(C_{\infty})=A<b>$.

\end{proposition}

Let's note, if the norm $N_{G}(A_{\infty})$ is non-Abelian in non-periodic
group, then the norm $N_{G}(C_{\infty})$ of infinite cyclic subgroups is
non-Abelian. Moreover, in this case $N_{G}(C_{\infty})= N_{G}(A_{\infty})$. The
following example shows that non-periodic groups, which norm
$N_{G}(C_{\infty})$ is non-Abelian and norm $N_{G}(A_{\infty})$ is Abelian, exist.

\begin{example} $G=(<a>\lambda <b>)\times C, |a|= \infty,
|b|=2, C$ is an infinite elementary Abelian 2-group, $b^{-1}ab=a^{-1}$.

It is easy to prove that $N_{G}(A_{\infty})=C$ is Abelian,
$N_{G}(C_{\infty})=G$ and $N_{G}(A_{\infty})\not=N_{G}(C_{\infty})$.
\end{example}

The following proposition characterizes the conditions when the norm
$N_{G}(A_{\infty})$ coinsides with the norm $N_{G}(C_{\infty})$ in non-periodic
group $G$ (provided that the subgroup $N_{G}(C_{\infty})$ is non-Abelian).

\begin{proposition} Let $G$ be a non-periodic group, which norm
$N_{G}(C_{\infty})$ of infinite cyclic subgroups is non-Abelian. Subgroups
$N_{G}(C_{\infty})$ and \\ $N_{G}(A_{\infty})$ coincide, if and only if
$N_{G}(C_{\infty})$ is central-by-finite and contains every infinite Abelian
subgroup of $G$.
\end{proposition}

In connection with the existence of O.~Yu.~Olshansky groups, periodic groups
with non-Dedekind norm of infinite Abelian subgroups were studied under the
condition of their locally finiteness. In \cite{Lym:6} it was proved that such
groups satisfy the minimal condition for subgroups, if and only if subgroup
$N_{G}(A_{\infty})$ satisfies this condition. Moreover, if $N_{G}
\left(A_{\infty}\right)$ is a group with minimal condition for subgroups, then
$G$ is a finite extension of its divisible part and therefore $\left[G:N_{G}
\left(A_{\infty }\right)\right]<\infty$.

Note that the norm $N(G)$ can be considered as the intersection of the
normalizers of all cyclic subgroups. In this connection it is naturally to
consider $\Sigma$-norm, where $\Sigma$ consists of all cyclic subgroups of
non-prime order of this group. Such norm was studied by T.~D.~Lukashova and
M.~G.~Drushlyak \cite{Luk:5} in the class of non-periodic groups and was called
\textit{the norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order of the
group $G$}.

It is clear that all cyclic subgroups of compound or infinite order are normal
in non-periodic group $G$, which coincides with the norm $N_{G}(C_{\bar{p}})$.
Such non-Dedekind groups were studied by T.~G.~Lelechenko, F.~M.~Lyman
\cite{Lel:1} and were called \textit{almost Dedekind groups}.

Since the norm $N_{G}(C_{\bar{p}} )$ normalizes each infinite cyclic subgroup
of a group \textit{G}, $N_{G}(C_{\bar{p}})\subseteq N_{G}(C_{\infty })$ in
non-periodic groups It turns out that these norms coincide, if the norm $N_{G}
(C_{\bar{p}})$ of cyclic subgroups of non-prime order is non-Abelian.

\begin{proposition} (\cite{Luk:5}) Any non-periodic group $G$ that has
non-Abelian norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order is
almost Dedekind and coincides with this norm. Moreover $G=A\lambda \left\langle
b\right\rangle $, where $A$ is a normal Abelian subgroup which contains all
elements of prime order of group $G$, $\left|b\right|=2$, $b^{-1}ab=a^{-1}$ for
an arbitrary element $a\in A$.
\end{proposition}

In 2004 in \cite{Lym:9, Luk:6} F.~M.~Lyman and T.~D.~Lukashova introduced one
more $\Sigma $-norm, where $\Sigma$ is a system of all Abelian non-cyclic
subgroups of a group. This $\Sigma$-norm was called \textit{the norm of Abelian
non-cyclic subgroups} of a group $G$ and denoted by $N_{G}^{A}$. It is clear if
the group $G$ coincides with the norm $N_{G}^{A}$ then all Abelian non-cyclic
subgroups are normal in it (assuming the existence of at least one of such
subgroup). Non-Abelian groups with this property were fully described in
\cite{Lym:9,Lym:10, Lym:11} and called $\overline{HA}$-groups ($\overline{HA}_{p}
$-groups in the case of $p$-groups).

In \cite {Luk:1, Lym:9, Lym:14} infinite locally finite $p-$groups ($p$ is an
arbitrary prime), which norm $N_{G}^{A}$ is non-Dedekind, are considered. The
authors obtained a complete description of such groups and proved that if the
norm $N_{G}^{A}$ is infinite and non-Dedekind, then all Abelian non-cyclic
subgroups are normal in a group, that is in this case $G=N_{G}^{A}$. It was
also proved that locally finite $p$-groups with non-Dedekind norm $N_{G}^{A}$
are finite extensions of a quazicyclic group. In particular, following
propositions take place.

\begin{proposition} (\cite {Lym:9}) Infinite 2-groups with
non-Dedekind norm $N_{G}^{A} $ of Abelian non-cyclic subgroups are groups of one
of the following types:

\textit{1)} $G$ is infinite $\overline{HA}_{2}$-group, $N_{G}^{A}=G$;

\textit{2)} $G=(A\times<b>)\lambda<c>\lambda<d>$, where $A$ is quazicyclic
2-group,$|b|=|c|=|d|=2, [A,<c>]=1, [b,c]=[b,d]=[c,d]=a_{1}\in A, |a_{1}|=2,
d^{-1}ad=a^{-1}$ for an arbitrary element $a\in A$; $N_{G}^{A}=N_{G}
=(<a_{2}>\times<b>)\lambda<c>, a_{2}\in A, |a_{2}|=4$;

\textit{3)} $G=A<y>H$, where $A$ is quazicyclic 2-group, $[A,H]=E,
H=<h_{1},h_{2}>, |h_{1}|=4, h_{1}^{2}=h_{2}^{2}=[h_{1},h_{2}]$, $|y|=4,
y^{2}=a_{1}\in A, y^{-1}ay=a^{-1}$ for an arbitrary element $a\in
A,[<y>,H]\subseteq <a_{1}>\times <h^{2}>$; $N_{G}^{A} =<a_{2}>\times H,
a_{2}\in A, |a_{2}|=4, N_{G}=\left\langle h_{2}\right\rangle\lambda
\left\langle h_{1}a_{2}\right\rangle$.

\end{proposition}

\begin {proposition} (\cite{Luk:6}) Infinite locally finite $p$-groups
($p\neq 2$), which norm $N_{G}^{A}$ of Abelian non-cyclic subgroups is
non-Dedekind, are $\overline{HA_{p}}$-groups and $G=N_{G}^{A}=N_{G}$.
\end{proposition}

Let's note that Proposition 4.14 fails in the case of infinite locally
finite 2-groups: there are infinite 2-groups with finite non-Dedekind norm
$N_{G}^{A}$ of Abelian non-cyclic subgroups, which may not coincide with the
norm $N_{G}$.

The study of finite $p$-groups for certain restrictions on the norm $N_{G}^{A}$
of Abelian non-cyclic subgroups were continued by M.~G.~Drushlyak,
T.~D.~Lukashova and F.~M.~Lyman in \cite {Dru:1,Lym:13}. In particular, in
\cite{Dru:1} the structure of finite $p$-groups ($p\ne 2$) with non-Abelian
norm of Abelian non-cyclic subgroups was completely described, in \cite{Lym:13}
the structure of finite 2-groups with non-cyclic centre and non-Dedekind norm
$N_{G}^{A}$ of Abelian non-cyclic subgroups was described. It is also proved
that an arbitrary 2-group with non-cyclic centre and non-Dedekind norm
$N_{G}^{A}$ does not contain quaternion subgroup, if and only if the norm
$N_{G}^{A}$ does not contain such subgroup. In this case the norm $N_{G}^{A}$
coinsides with the norm $N_{G}$ \cite {Lym:13}.

The following proposition clarifies the result of \cite{Luk:1} on the
coincidence of norms $N_{G}^{A}$ and $N_{G}$ for infinite locally finite
$p$-groups ($p\ne 2$) under the condition that the subgroup $N_{G}$ is
non-Abelian.

\begin{proposition} (\cite{Dru:1}) If either norm $N_{G}^{A}$ or
$N_{G}$ is non-Abelian, then $N_{G}=N_{G}^{A}$ in the class of locally finite
$p$-groups ($p\ne 2$).
\end{proposition}

The last proposition leads to the conclusion that any finite $p$-group ($p\ne
2$) with non-Abelian norm $N_{G}^{A}$ is a group of one of the types of 1)-3)
of Proposition 4.2.

\begin{proposition} (\cite{Lym:13}) Finite 2-groups with non-cyclic centre
and non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups is a group of
the following types:

\textit{1)} $G$ is non-Dedekind non-metacyclic $\overline{HA_{2}}$--group with
non-cyclic center, $G=N_{G}^{A}$;

\textit{2)} $G=H\cdot Q$ is the product of quaternion group of order 8 and
generalized quaternion group; $H=\left\langle h_{1},h_{2} \right\rangle $,
$\left|h_{1} \right|=\left|h_{2}\right|=4$, $\left[h_{1},h_{2}
\right]=h_{1}^{2}=h_{2}^{2}$, $Q=\left\langle y,x\right\rangle$,
$\left|y\right|=2^{n}$, $n\ge 3, |x|=4, y^{2^{n-1}}=x^{2}, x^{-1}yx=y^{-1}, \left[Q,H\right]\subseteq
\left\langle x^{2},h_{1}^{2}\right\rangle; N_{G}^{A}=\left\langle y^{2^{n-2}}
\right\rangle \times H$;

\textit{3)} $G=\left\langle x\right\rangle \left\langle b\right\rangle,
\left|x\right|=2^{k}$, $\left|b\right|=2^{m}, m>2, k\ge m+r, 1\le r<m-1,
Z(G)=\left\langle x^{2^{r+1}}\right\rangle \times \left\langle b^{2^{r+1}}
\right\rangle, \left[x,b\right]=x^{2^{k-r-1}s} b^{2^{m-1}t},
\left(s,2\right)=1, 0\le t<2; N_{G}^{A}=N_{G}=\left\langle x^{2^{m-1}}
\right\rangle \lambda \left\langle b\right\rangle$.
\end{proposition}

Continuing study of finite 2-groups T.~D.~Lukashova, F.~M.~Lyman and
M.~G.~Drushlyak obtained structural description of groups with cyclic center
and non-metacyclic non-Dedekind norm \textit{$N_{G}^{A}$}.

\begin{proposition} Finite 2-groups with non-metacyclic
non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups and cyclic centre
are groups of the following types:

\textit{1)} $G$ is non-metacyclic non-Hamiltonian $\overline{HA}_{2}$-group
with cyclic center, $G=N_{G}^{A}$;

\textit{2)} $G=\left(\left\langle x\right\rangle \lambda \left\langle
c\right\rangle \right)\lambda \left\langle b\right\rangle $,
$\left|x\right|=2^{n}$, $n>3$, $\left|b\right|=\left|c\right|=2$,
$\left[x,b\right]=x^{\pm 2^{n-2}}c$,
$\left[b,c\right]=\left[x,c\right]=x^{2^{n-1}}$, $N_{G}^{A}=N_{G}
=(\left\langle x^{2}\right\rangle \times \left\langle c\right\rangle )\lambda
\left\langle b\right\rangle$;

\textit{3)} $G=(\left\langle x\right\rangle \times \left\langle b\right\rangle
)\lambda \left\langle c\right\rangle \lambda \left\langle d\right\rangle $,
$\left|x\right|=2^{n}, n>2, \left|b\right|=\left|c\right|=\left|d\right|=2$,
$\left[x,c\right]=\left[x,b\right]=1$,
$\left[b,c\right]=\left[c,d\right]=\left[b,d\right]=x^{2^{n-1}}$, $d^{-1}
xd=x^{-1}$, $N_{G}^{A}=N_{G}=(\left\langle x^{2^{n-2}}\right\rangle \times
\left\langle b\right\rangle )\lambda \left\langle c\right\rangle$;

\textit{4)} $G=\left(\left\langle c\right\rangle \lambda H\right)\left\langle
y\right\rangle$, $H=\left\langle h_{1},h_{2} \right\rangle, \left|h_{1}
\right|=\left|h_{2}\right|=4, h_{1}^{2} =h_{2}^{2}=\left[h_{1},h_{2}\right]$,
$\left|c\right|=4, \left[c,h_{2} \right]=1,$ $\left[c,h_{1}\right]=c^{2},$
$y^{2}=h_{1},\left[y,h_{2}\right]=c^{2}h_{1}^{2},\left[y,c\right]=h_{2}^{\pm
1}; N_{G}^{A} =\left\langle c\right\rangle \lambda H$.
\end{proposition}

The question of the structure of finite 2-groups with cyclic center, in which
the norm $N_{G}^{A}$ is metacyclic non-Dedekind group, is still open.

Study of the impact of the properties of the norm of Abelian non-cyclic
subgroups on the properties of the group was continued in \cite{Lym:14}, where
infinite periodic groups, which norm $N_{G}^{A}$ is non-Dedekind and locally
nilpotent, were considered. It was proved that such groups satisfy the minimal
condition for Abelian subgroups and are Chernikov groups.

\begin{proposition} (\cite{Lym:14}) Infinite periodic locally nilpotent
group $G$ has non-Dedekind norm of Abelian non-cyclic subgroups, if and only if
$$G=G_{p}\times G_{p'},$$ where $G_{p}$ is infinite Sylow $p$-subgroup of a group
$G$ with non-Dedekind norm $N_{G_{p}}^{A}$ of Abelian non-cyclic subgroups
(where $p\in\pi(G)$) and $G_{p'}$ is finite cyclic or finite Hamiltonian
$p'$-subgroup, all Abelian subgroups of which are cyclic, and $N_{G}^{A}
=N_{G_{p}}^{A}\times G_{p'}$.
\end{proposition}

If $G$ is locally finite, not locally nilpotent group which has infinite locally
nilpotent non-Dedekind norm $N_{G}^{A}$, then $G=G_{p}\leftthreetimes H$, where
$G_{p}$ is infinite $\overline{HA}_{p}$-group, which coincides with Sylow
$p$-subgroup of norm $N_{G}^{A}$, and $H$ is a finite group, all Abelian
subgroups of which are cyclic, $\left(\left|H\right|,p\right)=1$. In
addition, the structure of infinite locally finite non-nilpotent groups, which
norm $N_{G}^{A}$ is finite non-Dedekind nilpotent subgroup, was described.

Study of the norm $N_{G}^{A}$ of Abelian non-cyclic subgroups in the class of
non-periodic groups were continued by M.~G~Drushlyak and F.~M.~Lyman. In
\cite{Dru:2, Lym:15} non-periodic groups with non-Dedekind norm of Abelian
non-cyclic subgroups depending on the presence \cite{Dru:2} or the absence
\cite{Lym:15} of free Abelian subgroup of rank 2 were considered.

\begin{proposition} (\cite{Dru:2}) If non-periodic group $G$ contains an free
Abelian subgroup of rank 2, its norm $N_{G}^{A}$ of Abelian non-cyclic
subgroups is non-Dedekind, contains an Abelian non-cyclic subgroup and a finite
Abelian, normal in $G$, subgroup $F$ and the centralizer $C_{G}(F)$ contains all
elements of infinite order of a group, then $N_{G}^{A}=N_{G}(C_{\infty}
)=B\left\langle d\right\rangle$, where $B$ is the Abelian subgroup generated by
all elements of infinite order of the group $G$, $\left|d\right|=2$ or
$\left|d\right|=4$, $d^{2}\in B$, $d^{2}$ is a unique involution in $G$ and
$d^{-1} bd=b^{-1}$ for an arbitrary element $b\in B$.
\end{proposition}

It was also proved that the non-periodic group $G$ does not contain a free
Abelian subgroups of rank 2, if its norm $N_{G}^{A}$ is non-Hamiltonian
$\overline{HA}$-group and does not contain such subgroups.

In 2015 F.~M.~Lyman and T.~D.~Lukashova \cite{Lym:16} considered one more
generalization of the concept of the norm of the group -- \textit{the norm
$N_{G}^{d}$ of decomposable subgroups}, which is defined as the intersection of
the normalizers of all decomposable subgroups of the group. In the case when
the group does not contain any decomposable subgroups, we can assume that
$N_{G}^{d} =G$. The structure of locally soluble groups, in which system of
decomposable subgroups is empty, as well as groups, in which each decomposable
subgroup is normal (groups with the condition $N_{G}^{d}=G$), was described in
\cite{Lym:17}.

It is clear that the group contains decomposable subgroups, if and only if it
contains decomposable Abelian subgroups. Therefore, the study of the norm $N_{G}^{d}$ of
decomposable subgroups was conducted, depending on the existence of systems of
decomposable Abelian subgroups in the group. Thus the norm $N_{G}^{d}$ of
decomposable subgroups is closely related to the norm $N_{G}^{A}$ of Abelian
non-cyclic subgroups. In particular, in \cite{Lym:16} it was proved that these
norms coinside in the class of locally finite $p$-groups. The inclusion
$N_{G}^{A} \supseteq N_{G}^{d}$ takes place and the case $N_{G}^{A} \ne
N_{G}^{d}$ is achieved in classes of finite non-primary groups, as well as in
classes of infinite periodic locally nilpotent non-primary groups.

\begin{proposition} (\cite{Lym:16}) Periodic locally nilpotent group $G$
which contains an Abelian non-cyclic subgroup has non-Dedekind norm $N_{G}^{d}
$ of decomposable subgroups, if and only if $G$ is a locally finite $p$-group
with non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups.
\end{proposition}

Study of the impact of the norm $N_{G}^{d}$ of decomposable subgroups on the properties of the
group were extended by the authors in the class of non-periodic groups. In
particular, in \cite{Lym:18} the following was established.

\begin{proposition}\cite{Lym:18} Let $G$ be a non-periodic group  has
non-Dedekind norm $N_{G}^{d}$ of decomposable subgroups. Then the following
propositions take place:

\textit{1)} the group $G$ does not contain decomposable subgroups if and only
if the norm $N_{G}^{d}$ of the group does not contain them;

\textit{2)} the group $G$ contains a free Abelian subgroup of rank $r\geq 2$,
if and only if the norm $N_{G}^{d}$ contains free Abelian subgroup of such
rank;

\textit{3)} the group $G$ contains non-primary Abelian subgroup, if and only if
the norm $N_{G}^{d}$ of the group contains subgroups with this property;

\textit{4)} any decomposable Abelian subgroup of a group G is mixed, if and only
if any decomposable Abelian subgroup of its norm $N_{G}^{d}$ is mixed.
\end{proposition}

It was also proved that in the class of non-periodic locally soluble groups
only one of the inclusions $N_{G}^{A}\supseteq N_{G}^{d}$ or $N_{G}^{A}
\subseteq N_{G}^{d}$ takes place, provided that at least one of these norms is non-Dedekind
and the norm $N_{G}^{d}$ is infinite. The following examples confirm that the
condition of the infiniteness of the norm $N_{G}^{d}$ is essential.

\begin{example}\cite{Lym:18} $G=\left(\left\langle a\right\rangle \leftthreetimes
B\right)\leftthreetimes \left\langle c\right\rangle$, where $|a|=p, p$ is a
prime ($p\neq 2$), $B$ is the group isomorphic to the additive group of $q$-adic
numbers, $q\notin \left\{2,p\right\}, B=B_1\left\langle x\right\rangle, x^2\in
B_1,$ $x^{-1}ax=a^{-1}, [B_1,\left\langle a\right\rangle]=E, |c|=2,
\left[c,a\right]=1, c^{-1}bc=b^{-1}$ for an arbitrary element $b\in B$.
\end{example}

In this group all periodic decomposable subgroups are of order $2p$ and are
groups of the type $\left\langle a^mcb^k_1\right\rangle$, where $b_1{\in B}_1,\
k\in \left\{0,1\right\},\left(m,p\right)=1$. Accordingly, all non-periodic
decomposable subgroups are mixed and contained in the group $B_1\times
\left\langle a\right\rangle$ and therefore are normal in $G$. Since
$N_G\left(\left\langle a^mcb^k_1\right\rangle \right)=\left\langle
a^mcb^k_1\right\rangle $, $N^d_G=\left\langle a\right\rangle.$

On the other hand, $G$ does not contain periodic Abelian non-cyclic subgroups
and all mixed Abelian subgroups contain $\left\langle a\right\rangle$ and are
subgroups of the group $\left(B_1\times \left\langle a\right\rangle \right)$
and therefore are normal in $G$. Moreover, all Abelian non-cyclic subgroups of
rank 1 are contained either in the subgroup $B$, or in subgroups conjugated
with it $g^{-1}Bg, g\in G$, or in the group $\left(B_{1}\times \left\langle
a\right\rangle \right)$.

Let's consider an infinite sequence of subgroups in $B_1$:

\centerline {$\left\langle b_1\right\rangle \subset \left\langle
b_2\right\rangle \subset \dots \left\langle b_n\right\rangle \subset...,$}

\noindent $\left|b_1\right|=\infty, b^{\alpha_{n+1}}_{n+1}=b_n$,
$\alpha_{n+1}\in {\mathbb N}, \left(\alpha_{n+1},p\right)=1$ for $n=1,2,....$
Since the isolator $A$ of the subgroup $\left\langle ab_1\right\rangle $ is
non-cyclic (because the root of arbitrary degree coprime with $p$ can be taken
from the element $a$), $N_G\left(A\right)=\left\langle {a,B}_1\right\rangle$.
$N^A_G=B_1$ and $N^d_G\cap N^A_G=E$ by $N_G\left(B\right)=B\leftthreetimes
\left\langle c\right\rangle $.

\begin{example}\cite{Lym:18} $G=\left(\left\langle a\right\rangle \leftthreetimes
B\right)\leftthreetimes \left\langle c\right\rangle$, where $|a|=p$, $p$ is a prime
($p\neq 2$), $B$ is the group isomorphic to the additive group of $p$-adic numbers,
$B=B_1\left\langle x\right\rangle, x^2\in B_1,
x^{-1}ax=a^{-1},\left[B_1,\left\langle a\right\rangle \right]=E,
\left|c\right|=2,\left[c,a\right]=1, c^{-1}bc=b^{-1}$ for an arbitrary element
$b\in B$.
\end{example}

As in example 4.2 in this group the norm of decomposable subgroups is
$N^d_G=\left\langle a\right\rangle.$ However, the norm of Abelian non-cyclic
subgroups is $N^A_G=\left(B_1\leftthreetimes \left\langle c\right\rangle
\right)$. This follows from the fact that for an arbitrary not unit element
$y_1\in B_1$ the isolator of subgroup $\left\langle ay_1\right\rangle $ is
cyclic, and therefore the element $c$ normalizes each Abelian non-cyclic
subgroup of a group $G$. In this case, the norm $N^A_G$ of Abelian non-cyclic
subgroups is non-Dedekind and $N^d_G\cap N^A_G=E.$

In 2005 F.~Mari, F.~de~Giovanni \cite{Mari:1} considered the concept of
\textit{the non-Abelian norm $N^{*}(G)$} that is the intersection of
normalizers of all non-Abelian subgroups of the group. If $N^{*}(G)=G$, then
all non-Abelian subgroups are normal in the group. These groups were studied by
G.~M.~Romalis and N.~F.~Sesekin \cite{Rom:1,Ses:1,Rom:2} and were called
\textit{metahamiltonian}. Further metahamiltonian groups were studied by
V.~T.~Nagrebezkiy \cite{Nag:1}, O.~A.~Makhnev \cite{Makh:1}, S.~M.~Chernikov
\cite{Chern:3}, M.~M.~Semko and M.~F.~Kuzennyi \cite{Kuz:1}.

In \cite{Mari:1} the results that generalize Schur theorem \cite{Gor:1} on
finiteness of derived subgroups in central-by-finite groups were got.

\begin{proposition} (\cite{Mari:1}) If $G$ is a locally graded group and
the quotient group $G/N^{*}(G)$ is finite, then derived subgroup $G'$ is
finite.
\end{proposition}

\section*{Conclusion} Authors can conclude that the study of different
$\Sigma$-norms and properties of groups with respect on properties of their
$\Sigma$-norms is very important area in the group theory. Nowadays research of
groups that differ from their $\Sigma $-norms as well as groups that have
non-Dedekind $\Sigma $-norm becomes possible, becourse the structure of groups
that coincide with $\Sigma $-norms is well known in many cases. This will give
the opportunity to extend the known classes of generalized Dedekind groups and
will allow to study groups with restrictions on the normalizers of different
systems of subgroups more effective.

There are a number of problems in the study of groups with generalized norms:

-- the study of groups that coincide with their $\Sigma $-norms;

-- the study of groups that have trivial $\Sigma $-norms or their $\Sigma
$-norms coinside with the center;

-- the study of groups that have non-central Dedekind $\Sigma $-norms;

-- the study of groups that have non-Dedekind $\Sigma $-norms;

-- the study of infinite groups that have $\Sigma $-norms of finite index.

The solution of these problems will significantly expand the base of the modern
group theory.

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\bibitem{Makh:1} A.~A.~Makhnev, \emph{On finite metahamiltonian groups}, Math. Notes
of Ural University, N.\textbf{1}, 1976, pp.60-75.

\bibitem{Chern:3} S.~N.~Chernikov, \emph{Groups with given properties of system of
subgroups,} M.: Nauka, 1980.

\bibitem{Kuz:1} M.~F.~Kuzennyi, M.~M.~Semko, \emph{Metahamiltonian groups
and their generalizations,} K: Math. Inst. NAS Ukraine, 1996.

\bibitem{Gor:1} Yu.~M.~Gorchakov, \emph{Groups with finite classes of conjugate
elements,} M.: Nauka, 1978.


\end{thebibliography}

\end{document}


\begin{proposition}
Use \verb"\label" and \verb"\ref" commands for referrings to the paper parts.
See source of the lemma~\ref{l:1} and sections~\ref{s:2},\ref{s:3}.
\end{proposition}


\begin{corollary}
Use \verb"thebibliography" environment for your paper
bibliographies. And the \verb"\cite" command for the bibliography
references.
\end{corollary}

\begin{proposition}
Use \texttt{proof} environment for the proofs. See proof of the
lemma~\ref{l:1}.
\end{proposition}

\begin{lemma}\label{l:1}
This is lemma 1.
\end{lemma}

\begin{proof}
This is proof of the lemma 1.
\end{proof}

\begin{theorem}
This is reference to the lemma~\ref{l:1}.
\end{theorem}

\section{Second section}\label{s:2}
\begin{example}
This is reference to bibliography~\cite{texbook}.
\begin{remark}
Form of references to the ADM  paper see in the bibliography
item~\cite{adm-journal}
\end{remark}
\end{example}

\section{Third section}\label{s:3}
\begin{proposition}
For more information about \TeX and \LaTeX2e typesetting
read~\cite{texbook},~\cite{latex}.
\end{proposition}

\section*{Conclusion}
For any questions about using this template to preparing ADM paper
and found bugs please contact to ADM editorial board.

\begin{thebibliography}{3}
\bibitem{texbook}
Donald~E. Knuth, \emph{The {{\TeX}book}}, Addison-Wesley, 1984.
\bibitem{latex}
Leslie Lamport, \emph{{\LaTeX:} {A} document preparation system},
  Addison-Wesley, 1986.
\bibitem{adm-journal} F. Author, S. Author, \emph{Paper title}, Algebra discrete
math., N.\textbf{1}, 2003, pp.1-2.
\end{thebibliography}

\end{document}
