16.54 (2006)
SolvedWe say that a group $G$ acts freely on a group $V$ if $vg \neq v$ for any nontrivial elements $g \in G$, $v \in V$. Is it true that a group $G$ that can act freely on a non-trivial abelian group is embeddable in the multiplicative group of some skew-field?
Progress
No, it is not. For example, the group $2.A_5.2$ with a quaternion Sylow 2-subgroup can act freely on an elementary abelian group of order $7^4$ but is not embeddable in the multiplicative group of any skew-field by Theorem 7 in (S. A. Amitsur, Trans. Amer. Math. Soc., 80, no. 2 (1955), 361–386). (D. Nedrenko, Letter of 20 January 2014.)
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