The Baer Invariant of Semidirect and Verbal Wreath Products of Groups

TL;DR

This paper extends formulas for the Baer invariant of semidirect and verbal wreath products of groups, generalizing previous results and providing new structural insights, especially for cyclic groups and free wreath products.

Contribution

It introduces new formulas and structural results for the Baer invariant of semidirect and verbal wreath products with respect to nilpotent varieties, generalizing Haebich's work.

Findings

${ m V}M(B)$ and ${ m V}M(A)$ are direct factors of ${ m V}M(G)$

Formulas of Haebich's type for ${ m N}_cM(B hd<A)$ when $B$ and $A$ are cyclic

Structural description of the Baer invariant for free wreath products with cyclic base groups

Abstract

W. Haebich (1977, Journal of Algebra {\bf 44}, 420-433) presented some formulas for the Schur multiplier of a semidirect product and also a verbal wreath product of two groups. The author (1997, Indag. Math., (N.S.), {\bf 8}({\bf 4}), 529-535) generalized a theorem of W. Haebich to the Baer invariant of a semidirect product of two groups with respect to the variety of nilpotent groups of class at most $c\geq 1,\ {\cal N}_c$. In this paper, first, it is shown that ${\cal V}M(B)$ and ${\cal V}M(A)$ are direct factors of ${\cal V}M(G)$, where $G=B\rhd<A$ is the semidirect product of a normal subgroup $A$ and a subgroup $B$ and $\cal V$ is an arbitrary variety. Second, it is proved that ${\cal N}_cM(B\rhd<A)$ has some homomorphic images of Haebich's type. Also some formulas of Haebich's type is given for ${\cal N}_cM(B\rhd<A)$, when $B$ and $A$ are cyclic groups. Third, we will present a formula for the Baer invariant of a $\cal V$-verbal wreath product of two groups with respect to the variety of nilpotent groups of class at most $c\geq 1$, where $\cal V$ is an arbitrary variety. Moreover, it is tried to improve this formula, when $G=A{\it Wr_V}B$ and $B$ is cyclic. Finally, a structure for the Baer invariant of a free wreath product with respect to ${\cal N}_c$ will be presented, specially for the free wreath product $A{\it Wr_*}B$ where $B$ is a cyclic group.