Geodesic
Primary cosets in groups
Algebra i logika, Tome 59 (2020) no. 3, pp. 315-322.

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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.
Keywords: generalized Frobenius group, projective special linear group, coset.
Mots-clés : insoluble group 
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A. Kh. Zhurtov; D. V. Lytkina; V. D. Mazurov. Primary cosets in groups. Algebra i logika, Tome 59 (2020) no. 3, pp. 315-322. http://geodesic.mathdoc.fr/item/AL_2020_59_3_a2/

[1] M. T. Lewis, “Camina groups, Camina pairs, and generalization”, Group theory and computation, Invited papers based on the presentations at the workshop "‘Group theory and computational methods"’ (Bangalore, India, November 5–14, 2016), Indian Stat. Inst. Ser., eds. N. S. N. Sastry, M. K. Yadav, Springer, Singapore, 2018, 41–174 | MR

[2] S. Vei, A. Kh. Zhurtov, D. V. Lytkina, V. D. Mazurov, “Konechnye gruppy, blizkie k gruppam Frobeniusa”, Sib. matem. zh., 60:5 (2019), 1035–1040 | MR

[3] X. Wei, W. B. Guo, D. V. Lytkina, V. D. Mazurov, A. Kh. Zhurtov, “Solubility of finite generalized Frobenius groups with the kernel of odd index”, Izv. NAN Armenii: Matem., 55:1 (2020), 88–92 | MR | Zbl

[4] W. Burnside, “On an unsettled question in the theory of discontinuous groups”, Q. J. Pure Appl. Math., 33 (1902), 230–238 | Zbl

[5] W. Burnside, “On groups in which every two conjugate operations are permutable”, Proc. Lond. Math. Soc., 35 (1903), 28–37 | MR | Zbl

[6] C. Hopkins, “Finite groups in which conjugate operations are commutative”, Am. J. Math., 51 (1929), 35–41 | DOI | MR | Zbl

[7] F. Levi, B. L. van der Waerden, “Über eine besondere Klasse von Gruppen”, Abh. Math. Semin. Univ. Hamb., 9 (1932), 157–158 | MR

[8] B. H. Neumann, “Groups with automorphisms that leave only the neutral element fixed”, Arch. Math., 7:1 (1956), 1–5 | DOI | MR | Zbl

[9] A. Kh. Zhurtov, “O regulyarnykh avtomorfizmakh poryadka 3 i parakh Frobeniusa”, Sib. matem. zh., 41:2 (2000), 329–338 | MR | Zbl

[10] C. H. Li, L. Wang, “Finite REA-groups are solvable”, J. Algebra, 522 (2019), 195–217 | DOI | MR | Zbl

[11] B. Huppert, Endliche Gruppen I, Grundlehren Math. Wiss., 134, Springer-Verlag, Berlin a.o., 1979 | MR | Zbl