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Automorphisms of Divisible Rigid Groups

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Algebra and Logic Aims and scope

A group G is m-rigid if there exists a normal series of the form G = G1 > G2 > . . . > Gm > Gm+1 = 1 in which every factor Gi/Gi+1 is an Abelian group and is torsion-free as a (right) Z[G/Gi]-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 Gi/Gi+1 are divisible modules over Z[G/Gi]. The rings Z[G/Gi] satisfy the Ore condition, and Q(G/Gi) denote the corresponding (right) division rings. Thus, for a divisible rigid group G, the factor Gi/Gi+1 can be treated as a (right) vector space over Q(G/Gi). 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.

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Authors and Affiliations

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

    D. V. Ovchinnikov

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  1. D. V. Ovchinnikov
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Correspondence to D. V. Ovchinnikov.

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Translated from Algebra i Logika, Vol. 53, No. 2, pp. 206-215, March-April, 2014.

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Ovchinnikov, D.V. Automorphisms of Divisible Rigid Groups. Algebra Logic 53, 133–139 (2014). https://doi.org/10.1007/s10469-014-9277-6

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  • DOI: https://doi.org/10.1007/s10469-014-9277-6

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