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Algebra Logika, 2014 Volume 53, Number 2, Pages 206–215 (Mi al631)

This article is cited in 1 paper

Automorphisms of divisible rigid groups

D. V. Ovchinnikov

Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

Abstract: A group $G$ is $m$-rigid if there exists a normal series of the form
$$ G=G_1>G_2>\ldots>G_m>G_{m+1}=1 $$
in which every factor $G_i/G_{i+1}$ is an Abelian group and is torsion-free as a (right) $\mathbb Z[G/G_i]$-module. A rigid group is one that is $m$-rigid for some $m$. The specified series is determined by a given rigid group uniquely; so it consists of characteristic subgroups and is called a rigid series; the solvability length of a group is exactly $m$. A rigid group $G$ is divisible if all $G_i/G_{i+1}$ are divisible modules over $\mathbb Z[G/G_i]$. The rings $\mathbb Z[G/G_i]$ satisfy the Ore condition, and $Q(G/G_i)$ denote the corresponding (right) division rings. Thus, for a divisible rigid group $G$, the factor $G_i/G_{i+1}$ can be treated as a (right) vector space over $Q(G/G_i)$.
We describe the group of all automorphisms of a divisible rigid group, and then a group of normal automorphisms. An automorphism is normal if it keeps all normal subgroups of the given group fixed.

Keywords: divisible rigid group, group of automorphisms, group of normal automorphisms.

UDC: 512.5

Received: 30.11.2013
Revised: 15.01.2014


 English version:
Algebra and Logic, 2014, 53:2, 133–139

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