This survey reports on recent progress made on finite subgroups of the unit group of integral group rings of finite groups. We show that the Gruenberg-Kegel graph of ZG coincides with that one of G provided |G| is divisible by at most three primes and give an outline how such a result may be obtained with the aid of computational algebra. In the last section we discuss this question for sporadic simple groups and their automorphism groups.
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