On nonabelian composition factors of a finite prime spectrum minimal group

Abstract

Suppose that L is a finite group, π(L) is the set of prime divisors of the order |L|, and Y is the class of finite groups G such that π(G) �= π(H) for any proper subgroup H of G. Groups from the class Y will be called prime spectrum minimal. Many but not all finite simple groups are prime spectrum minimal. For finite simple groups not from the class Y, the question whether they are isomorphic to nonabelian composition factors of groups from the class Y is interesting. We describe some finite simple groups that are not isomorphic to nonabelian composition factors of groups from the class Y and construct an example of a finite group from Y that has as its composition factor the finite simple sporadic McLaughlin group Mc L not from the class Y.