Profinite groups admitting just infinite quotients
An infinite group is called just infinite if all of its proper quotients are finite. There has been much interest in just infinite groups in recent years, particularly in the field of pro-\(p\) groups where people like to think of just infinite pro-\(p\) groups as playing a role in pro-\(p\) group theory similar to the role played by finite simple groups in finite group theory. Indeed the German term for just infinite is `unendlich einfach' (infinitely simple). This notion is enhanced by the fact that every finitely generated pro-\(p\) group has a just infinite quotient. This is proved using Zorn's Lemma and the Frattini-series [Proposition II.5 in Linear pro-\(p\)-groups of finite width, Lect. Notes Math. 1674, Springer, \textit{G. Klaas, C. R. Leedham-Green} and \textit{W. Plesken} (1997; Zbl 0901.20013)]. In this paper the author addresses the question of which profinite groups admit just infinite quotients. He proves that any profinite group whose order (as a supernatural number) is divisible only by finitely many primes admits just infinite quotients. He also proves that the property of admitting a just infinite quotient is preserved when going to open subgroups and finite extensions. This is used to deduce that every virtually projective group (not necessarily finitely generated) admits a just infinite quotient. There are other results concerning profinite groups acting on trees. Instead of using the Frattini subgroup the proofs concentrate on the intersection of maximal normal subgroups.
scientific article; zbMATH DE number 1753105
scientific article