On the Finite Prime Spectrum Minimal Groups

Let G be a finite group. The set of all prime divisors of the order of G is called the prime spectrum of G and is denoted by pi(G). A group G is called prime spectrum minimal if pi(G) not equal pi(H) for any proper subgroup H of G. We prove that every prime spectrum minimal group all of whose nonabelian composition factors are isomorphic to the groups from the set {PSL2(7), PSL2(11), PSL5(2)} is generated by two conjugate elements. Thus, we extend the corresponding result for finite groups with Hall maximal subgroups. Moreover, we study the normal structure of a finite prime spectrum minimal group with a nonabelian composition factor whose order is divisible by exactly three different primes.