Primary Cosets in Groups

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...

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Bibliographic Details
Published in:Algebra and logic 2020-07, Vol.59 (3), p.216-221
Main Authors: Zhurtov, A. Kh, Lytkina, D. V., Mazurov, V. D.
Format: Article
Language:English
Subjects:
Algebra
Fuzzy sets
Group theory
Kernels
Mathematical Logic and Foundations
Mathematical research
Mathematics
Mathematics and Statistics
Quotients
Set theory
Subgroups
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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.
ISSN:0002-5232
1573-8302
DOI:10.1007/s10469-020-09593-w