n-complete crossed modules and wreath products of groups

Year 2021, Volume: 10 Issue: 1, 38 - 45, 30.04.2021

M.a. Dehghani B. Davvaz

Abstract

In this paper we examine the $n$-completeness of a crossed module and we show that if $X=(W_1,W_2,\partial)$ is an $n$-complete crossed module, where $W_i=A_i wr B_i$ is the wreath product of groups $A_i$ and $B_i$, then $A_i$ is at most $n$-complete, for $i=1,2.$ Moreover, we show that when $X=(W_1,W_2,\partial)$ is an $n$-complete crossed module, where $A_i$ is nilpotent and $B_i$ is nilpotent of class $n$, for $i=1,2$, then if $A_i$ is an abelian group, then it is cyclic of order $p_i.$ Also, if $W_i=C_ pwr C_2$, where $p$ is prime with $p>3$, $i=1,2$, then $X=(W_1,W_2,\partial)$ is not an $n$-complete crossed module.

Keywords

crossed module, wreath products, commutator

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