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Transversals, commutators and solvability in finite groups. (English) Zbl 0837.20026

The present paper is a continuation of a paper by the author and T. Kepka [Bull. Lond. Math. Soc. 24, No. 4, 343-346 (1992; Zbl 0793.20064)]. Let \(G\) be a finite group and \(H\) a subgroup of \(G\). If \(A\) and \(B\) are two (left) cosets of \(H\) in \(G\) and \([A,B]\) is contained in \(H\), then \(A\) and \(B\) are defined to be \(H\)-connected. Theorem 2.1: If \(H\) is nilpotent, has two \(H\)-connected cosets, and the Sylow-2 subgroups of \(H\) are of at most class 2, then \(G\) is soluble. The problem is raised whether \(G\) is soluble without that constraint on the Sylow-2 subgroups of \(H\). Theorem 3.4: If \(H\) is a proper subgroup of \(G\) with \(H\)- connected transversals and has order 6, then \(G\) is soluble.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20F12 Commutator calculus
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20N05 Loops, quasigroups

Citations:

Zbl 0793.20064
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