International Journal of Mathematics, Game Theory and Algebra, Year (2006-5)

#### Title : THE BAER INVARIANT OF SEMIDIRECT AND VERBAL WREATH PRODUCTS OF GROUPS ( The Baer invariant of semidirect product and verbal wreath products of groups )

Authors: Behrooz Mashayekhy Fard ,

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#### 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.

#### Keywords

Baer invariant; semidirect product; verbal wreath product; nilpotent variety.
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@article{paperid:202168,
author = {Mashayekhy Fard, Behrooz},
title = {THE BAER INVARIANT OF SEMIDIRECT AND VERBAL WREATH PRODUCTS OF GROUPS},
journal = {International Journal of Mathematics, Game Theory and Algebra},
year = {2006},
month = {May},
issn = {1060-9881},
keywords = {Baer invariant; semidirect product; verbal wreath product; nilpotent variety.},
}