On the Shunkov groups acting freely on Abelian groups. (Q359347)

A group \(G\) is called a \textit{Shunkov group} if, for each finite subgroup \(F\) of \(G\), the subgroup generated by any two conjugate elements of prime order in the group \(N_G(F)/F\) is finite. With Theorem 1 the author proves that the set of elements of finite order in a Shunkov group of rank \(1\) (i.e. \(C_p\times C_p\)-free for all primes \(p\)) is a locally finite group. In Theorem 2 it is proved that the same conclusion holds for Shunkov groups acting regularly on Abelian groups (Theorem 2). These theorems generalize similar results by \textit{T. Grundhöfer} and \textit{E. Jabara} [Arch. Math. 97, No. 3, 219-223 (2011; Zbl 1241.20036)]. An interesting application of the above mentioned results is Corollary 1: If the group \(T_2(D)\) of affine transformations of a neardomain \(D\) is a Shunkov group and \(\mathrm{char}(D)\neq 2\), then \(T_2(D)\) has a locally finite periodic part and a regular elementary Abelian normal subgroup; furthermore \(D\) is a nearfield of finite characteristic.