This article is cited in 1 paper

Residual properties of automorphisms groups and split extensions

D. N. Azarov

Chair of Algebra and Mathematical Logic, Ivanovo State University, 39 Ermaka str., Ivanovo, 153025 Russia

Abstract: Let a group $G$ satisfy condition A: for every positive integer $n$ the number of all subgroups of the group $G$ of index $n$ is finite. We prove that if $G$ is virtually residually finite $p$-group for some prime $p$, then the automorphism group of the group $G$ is virtually residually finite $p$-group. A similar result is obtained for a split extension of the group $G$ by virtually residually finite $p$-group. Moreover, we prove that if the group $G$ is a virtually residually finite nilpotent $\pi$-group for some finite set $\pi$ of primes, then the automorphism group of the group $G$ and the split extension of the group $G$ by a virtually residually finite nilpotent $\pi$-group are virtually residually finite nilpotent $\pi$-groups.

Keywords: linear group, automorphism group, virtually residually finite $p$-group.

UDC: 512.543

Received: 17.02.2014

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2015, 59:8, 1–8

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