On Prime Spectrum of Maximal Subgroups in Finite Groups

Abstract

For a positive integer n, we denote by π(n) the set of all prime divisors of n. For a finite group G, the set [Formula: see text] is called the prime spectrum of G. Let [Formula: see text] mean that M is a maximal subgroup of G. We put [Formula: see text] and [Formula: see text]. In this notice, using well-known number-theoretical results, we present a number of examples to show that both K(G) and k(G) are unbounded in general. This implies that the problem “Are k(G) and K(G) bounded by some constant k?”, raised by Monakhov and Skiba in 2016, is solved in the negative.