Primary cosets in groups. (English. Russian original) Zbl 1484.20044

Zhurtov, A. Kh.; Lytkina, D. V.; Mazurov, V. D.

Algebra Logic 59, No. 3, 216-221 (2020); translation from Algebra Logika 59, No. 3, 315-322 (2020).

Summary: A finite group \(G\) is called a generalized Frobenius group with kernel \(F\) if \(F\) is a proper nontrivial normal subgroup of \(G\), and for every element \(Fx\) of prime order \(p\) in the quotient group \(G/F\), the coset \(Fx\) of \(G\) consists of \(p\)-elements. We study generalized Frobenius groups with an insoluble kernel \(F\). It is proved that \(F\) has a unique non-Abelian composition factor, and that this factor is isomorphic to \(L_2(3^{2^l})\) for some natural number \(l\). Moreover, we look at a (not necessarily finite) group generated by a coset of some subgroup consisting solely of elements of order three. It is shown that such a group contains a nilpotent normal subgroup of index three.

MSC:

20D99	Abstract finite groups
20E34	General structure theorems for groups

Keywords:

generalized Frobenius group; projective special linear group; insoluble group; coset

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