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This article is cited in 1 scientific paper (total 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.
Received: 30.11.2013 Revised: 15.01.2014
Citation:
D. V. Ovchinnikov, “Automorphisms of divisible rigid groups”, Algebra Logika, 53:2 (2014), 206–215; Algebra and Logic, 53:2 (2014), 133–139
Linking options:
https://www.mathnet.ru/eng/al631 https://www.mathnet.ru/eng/al/v53/i2/p206
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Abstract page: | 233 | Full-text PDF : | 62 | References: | 39 | First page: | 16 |
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