Niemenmaa, Markku Transversals, commutators and solvability in finite groups. (English) Zbl 0837.20026 Boll. Unione Mat. Ital., VII. Ser., A 9, No. 1, 203-208 (1995). 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. Reviewer: Hans Lausch (Clayton) Cited in 1 Document 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 Keywords:solubility; finite groups; cosets; \(H\)-connected cosets; Sylow-2 subgroups; \(H\)-connected transversals Citations:Zbl 0793.20064 PDFBibTeX XMLCite \textit{M. Niemenmaa}, Boll. Unione Mat. Ital., VII. Ser., A 9, No. 1, 203--208 (1995; Zbl 0837.20026)