On the Gruenberg–Kegel graph of integral group rings of finite groups

Abstract

The prime graph question asks whether the Gruenberg–Kegel graph of an integral group ring ℤG, i.e. the prime graph of the normalized unit group of ℤG, coincides with that one of the group G. In this note, we prove for finite groups G a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups G whose order is divisible by at most three primes and show that the Gruenberg–Kegel graph of such groups coincides with the prime graph of G.