## 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)].

### MSC:

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

### Keywords:

free actions; conjugate elements; Frobenius groups

### Citations:

Zbl 1009.20039; Zbl 0851.20039
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