On a group acting locally freely on an Abelian group. (Russian, English) Zbl 1035.20032

Sib. Mat. Zh. 44, No. 2, 343-346 (2003); translation in Sib. Math. J. 44, No. 2, 275-277 (2003).
A group \(G\) acts freely on a nontrivial Abelian group \(V\) if \(vg\neq v\) for all \(g\in G\), \(g\neq1\), and all \(v\in V\), \(v\neq 1\). The author uses the coset enumeration algorithm and calculations in the discrete computational algebra system GAP [M. Schönert et al. Groups, Algorithms and Programming (1997), http://www-gap.dcs.st-and.ac.uk/\(\sim\)gap] to prove the following Theorem: Let \(G\) be a group acting freely on an Abelian group \(V\) and let \(x\in G\) be an element of prime order \(p\). If the subgroup \(\langle x,x^g\rangle\) is finite and acts freely on \(V\) for all \(g\in G\) then the group \(H=\langle x^G\rangle\) is finite and acts freely on \(V\). More precisely, either \(| H|=p\), or \(H\simeq\text{SL}_2(3)\), or \(H\simeq\text{SL}_2(5)\).
This theorem generalizes some results obtained by V. D. Mazurov and V. A. Churkin [Sib. Mat. Zh. 43, No. 3, 600-608 (2002; Zbl 1009.20039)] and by A. I. Sozutov [Sib. Mat. Zh. 35, No. 4, 893-901 (1994; Zbl 0851.20039)].


20F28 Automorphism groups of groups
20D06 Simple groups: alternating groups and groups of Lie type
20F05 Generators, relations, and presentations of groups
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