This article is cited in 2 papers

On the second commutants of finite Alperin groups

B. M. Veretennikov

Ural Federal University, Ekaterinburg, Russia

Abstract: We refer to an Alperin group as a group in which the commutant of every $2$-generated subgroup is cyclic. Alperin proved that if $p$ is an odd prime then all finite $p$-groups with the property are metabelian. Nevertheless, finite Alperin $2$-groups may fail to be metabelian. We prove that for each finite abelian group $H$ there exists a finite Alperin group $G$ for which $G''$ is isomorphic to $H$.

Keywords: Alperin group, commutant (commutator subgroup), definition of a group by generators and defining relations.

UDC: 512.54

Received: 30.04.2013

 English version: Siberian Mathematical Journal, 2014, 55:1, 19–34

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