Bounds on the nilpotency class of certain semidirect products

Abstract:

Gilbert Baumslag has shown that the standard wreath product of $A$ by $B$ is nilpotent if and only if $A$ and $B$ are $p$-groups for the same prime $p, A$ is nilpotent of bounded exponent and $B$ is finite. L. Kaloujnine and Marc Krasner have shown that the standard (unrestricted) wreath product of $A$ by $B$ contains an isomorphic copy of every group $G$ which is an extension of $A$ by $B$. Thus it follows that any extension subject to the above condition on $A$ and $B$ is nilpotent. In this paper, the author gives an explicit characterization of the terms of the lower central series of a semidirect product $W$ of an abelian group by an arbitrary group. He then establishes a formula for an upper bound on the nilpotency class of $W$ when $W$ is a semidirect product of an abelian $p$-group $X$ of bounded exponent by a finite $p$-group $B$. This new bound is given in terms of the exponent of $X$ and the cycle structure of the factor groups of the lower central series of $B$.