The size of the commutator subgroup of finite groups.
Let \(G\) be a finite group. In the paper [J. Algebra 278, No. 2, 494-501 (2004; Zbl 1060.20018)] of the authors it was proved that if the group \(G\) is nontrivial solvable with trivial Frattini subgroup and with trivial center, then the order of the commutator subgroup is greater than the cube root of the order of \(G\). The aim of the paper is to prove that if the group \(G\) is nonabelian with trivial Frattini subgroup, then the order of the commutator subgroup is greater than the square root of the order of the central factor \(G/Z(G)\). The example of a Frobenius group of order \(p(p-1)\) with \(p\) a prime number shows that the inequality cannot be improved in general. -- The proof uses the \(k(GV)\)-theorem concerning the number of conjugacy classes of the semidirect product \(GV\) established recently by \textit{D. Gluck, K. Magaard, U. Riese, P. Schmid} [in J. Algebra 279, No. 2, 694-719 (2004; Zbl 1058.20010)].
scientific article; zbMATH DE number 5354918
scientific article