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On the Shunkov groups acting freely on Abelian groups. (English. Russian original) Zbl 1295.20039

Sib. Math. J. 54, No. 1, 144-151 (2013); translation from Sib. Mat. Zh. 54, No. 1, 188-198 (2013).
A group \(G\) is called a 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 T. Grundhöfer and 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.

MSC:

20F28 Automorphism groups of groups
20F50 Periodic groups; locally finite groups
20B22 Multiply transitive infinite groups
20E25 Local properties of groups
12K05 Near-fields

Citations:

Zbl 1241.20036
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References:

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