The Baer-invariant of a Semidirect Product

TL;DR

This paper generalizes a known result about the Schur-multiplier of a finite group, extending it to Baer-invariants of groups within the variety of nilpotent groups of class at most c, showing a direct factor relationship.

Contribution

It extends the theorem relating semidirect products and Schur-multiplier to Baer-invariants in the variety of nilpotent groups of class c.

Findings

${ m f N}_cM(T)$ is a direct factor of ${ m f N}_cM(G)$

Generalization from Schur-multiplier to Baer-invariant in nilpotent varieties

Applicable to arbitrary groups, not just finite groups

Abstract

In 1972 K.I.Tahara [7,2 Theorem 2.2.5], using cohomological method, showed that if a finite group $G=T\rhd<N$ is the semidirect product of a normal subgroup $N$ and a subgroup $T$, then $M(T)$ is a direct factor of $M(G)$, where $M(G)$ is the Schur-multiplicator of $G$ and in the finite case, is the second cohomology group of $G$. In 1977 W.Haebich [1 Theorem 1.7] gave another proof using a different method for an arbitrary group $G$ . In this paper we generalize the above theorem . We will show that ${\cal N}_cM(T)$ is a direct factor of ${\cal N}_cM(G)$, where ${\cal N}_c$ [3 page 102] is the variety of nilpotent groups of class at most $c\geq 1$ and ${\cal N}_cM(G)$ is {\it the Baer-invariant} of the group $G$ with respect to the variety ${\cal N}_c$ [3 page 107] .