The Baer Invariant of Semidirect and Verbal Wreath Products of 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.