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On finite Alperin 2-groups with elementary abelian second commutants

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Abstract

By an Alperin group we mean a group in which the commutant of each 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with this property are metabelian. The today’s actual problem is the construction of examples of nonmetabelian finite Alperin 2-groups. Note that the author had given some examples of finite Alperin 2-groups with second commutants isomorphic to Z 2 and Z 4 and proved the existence of finite Alperin 2-groups with cyclic second commutants of however large order by appropriate examples. In this article the existence is proved of finite Alperin 2-groups with abelian second commutants of however large rank.

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References

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Authors and Affiliations

  1. Ural Federal University, Ekaterinburg, Russia

    B. M. Veretennikov

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  1. B. M. Veretennikov
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Correspondence to B. M. Veretennikov.

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Dedicated to the blessed memory of my son Denis Veretennikov.

Original Russian Text Copyright © 2012 Veretennikov B. M.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 3, pp. 543–557, May–June, 2012.

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Veretennikov, B.M. On finite Alperin 2-groups with elementary abelian second commutants. Sib Math J 53, 431–443 (2012). https://doi.org/10.1134/S0037446612020243

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  • DOI: https://doi.org/10.1134/S0037446612020243

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