### A survey article on some subgroup embeddings and local properties for soluble PST-groups

#### Abstract

Let \(G\) be a group and \(p\) a prime number. \(G\) is said to be a \(Y_p\textrm{-group}\) if whenever \(K\) is a \(p\)-subgroup of \(G\) then every subgroup of \(K\) is an S-permutable subgroup in \(N_G(K)\). The group \(G\) is a soluble PST-group if and only if \(G\) is a \(Y_p\textrm{-group}\) for all primes \(p\).

One of our purposes here is to define a number of local properties related to \(Y_p\) which lead to several new characterizations of soluble PST-groups. Another purpose is to define several embedding subgroup properties which yield some new classes of soluble PST-groups. Such properties include weakly S-permutable subgroup, weakly semipermutable subgroup, and weakly seminormal subgroup.

#### Keywords

S-permutable subgroup, semipermutable subgroup, seminormal subgroup, PST-group

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