On Prime Spectrum of Maximal Subgroups in Finite Groups

For a positive integer n, we denote by ?n the set of all prime divisors of n. For a finite group G, the set ?G:= ?|G| is called the prime spectrum of G. Let M?G mean that M is a maximal subgroup of G. We put KG = max{|?G\?M |: M ? G} and kG = min{|?G\?M |: M ? G}. In this notice, using well-known number-theoretical results, we present a number of examples to show that both KG and kG are unbounded in general. This implies that the problem "Are kG and KG bounded by some constant k?", raised by Monakhov and Skiba in 2016, is solved in the negative.