The N‐prime graph and the Subgroup Isomorphism Problem

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