The N‐prime graph and the Subgroup Isomorphism Problem

Source abstract

Abstract We introduce a directed graph related to a group , which we call the N‐prime graph of and is a refinement of the classical Gruenberg–Kegel graph. The vertices of are the primes such that has an element of order , and, for distinct vertices and , the arc is in the graph if and only if has a subgroup of order whose normalizer in has an element of order . Generalizing some known results about the Gruenberg–Kegel graph, we prove that the group of the units with augmentation 1 in the integral group ring has the same N‐prime graph as if is a finite solvable group, and we reduce to almost simple groups the problem of whether holds for an arbitrary finite group . We also prove that if the finite group is almost simple with socle either an alternating group, or with prime and . Finally, for a finite solvable group we obtain some stronger results which give a contribution to the Subgroup Isomorphism Problem. More precisely, we prove that if contains a Frobenius subgroup with kernel of prime order and complement of prime‐power order, then contains a subgroup isomorphic to .