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 \( {\mathrm{L}}_2\left({3}^{2^{\mathrm{l}}}\right) \) 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.
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Translated from Algebra i Logika, Vol. 59, No. 3, pp. 315-322, May-June, 2020. Russian https://doi.org/10.33048/alglog.2020.59.302.
A. Kh. Zhurtov is Supported by RFBR, project No. 19-01-00507.
D. V. Lytkina is Supported by Mathematical Center in Akademgorodok, Agreement with RF Ministry of Education and Science No. 075-15-2019-1613.
V. D. Mazurov is Supported by SB RAS Fundamental Research Program I.1.1, project No. 0314-2019-001.
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Zhurtov, A.K., Lytkina, D.V. & Mazurov, V.D. Primary Cosets in Groups. Algebra Logic 59, 216–221 (2020). https://doi.org/10.1007/s10469-020-09593-w
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DOI: https://doi.org/10.1007/s10469-020-09593-w