On subgroups of finite exponent in groups

Abstract

We investigate properties of groups with subgroups of  finite exponent and prove that  a non-perfect group  \(G\)  of infinite exponent with all proper subgroups of finite exponent has the following properties:

\((1)\) \(G\) is an indecomposable  \(p\)-group,

\((2)\) if the derived subgroup \(G'\) is non-perfect, then \(G/G''\) is a group of Heineken-Mohamed type.

We also prove that  a non-perfect indecomposable group  \(G\) with the non-perfect locally nilpotent derived subgroup \(G'\)  is a locally finite \(p\)-group.