## 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

Zbl 0793.20064