A note on the solvability of a finite group and the number of conjugate classes of its non-normal subgroups.
For a group \(G\), denote by \(\nu(G)\) the number of conjugacy classes of non-normal subgroups of \(G\). The following result is announced: Every finite group \(G\) with \(\nu(G)\leq 6\) is soluble. The proof of this contains a number of gaps. For example, Theorem 1.1 deals with a finite group \(G\) whose Sylow 2-subgroup \(S_2\) contains at most two \(G\)-conjugacy classes of subgroups that are non-normal in \(G\). The authors claim that either \(G\) is soluble or all proper subgroups of \(S_2\) are normal in \(G\). A counterexample for this is the alternating group \(A_5\). The proof of the aforementioned solubility result relies on Theorem 1.1. Reviewer's remark: The aforementioned result, however, is correct. It follows from the classification of all nonsoluble finite groups \(G\) with \(\nu(G)<14\); see [\textit{R. Brandl}, Beitr. Algebra Geom. 54, No. 2, 493--501 (2013; Zbl 1284.20022)].
scientific article; zbMATH DE number 5951294