Subgroup Theorems for the Baer-invariant of Groups

TL;DR

This paper extends subgroup theorems for the Baer-invariant of groups, focusing on the center by a variety of groups, and derives new results related to Schur's theorem for nilpotent groups of specific classes.

Contribution

It proves a new subgroup theorem for the Baer-invariant involving the center and a variety of groups, generalizing previous results for the Schur multiplier.

Findings

Established a subgroup theorem for the Baer-invariant related to the center by a variety.

Derived Schur-type results for nilpotent groups of certain classes.

Extended known theorems to broader classes of groups and varieties.

Abstract

M.R.Jones and J.Wiegold in [3] have shown that if $G$ is a finite group with a subgroup $H$ of finite index $n$, then the $n$-th power of Schur multiplier of $G$, $M(G)^n$, is isomorphic to a subgroup of $M(H)$. In this paper we prove a similar result for the centre by centre by $w$ variety of groups, where $w$ is any outer commutator word. Then using a result of M.R.R.Moghaddam [6], we will be able to deduce a result of Schur's type (see [4,9]) with respect to the variety of nilpotent groups of class at most $c$ $(c\geq 1)$, when $c+1$ is any prime number or 4.