The hypercenter of a profinite group.
The hypercenter of a group \(G\) is usually defined as the union of the members of the upper central series. However, there are pronilpotent, non-trivial groups with trivial center. For a profinite group \(G\), it thus makes sense to define the (upper) hypercenter \(Z^\infty(G)\) as the set of elements that map into the hypercenter of the (finite) quotients modulo all open normal subgroups. The paper under review gives various characterizations of \(Z^\infty(G)\), as the largest subgroup normalizing every Sylow subgroup; as the largest pronilpotent normal subgroup \(Z\) such that \(G\) induces on each Sylow \(p\)-subgroup of \(Z\) a pro-\(p\) group of automorphisms, for all primes \(p\); as the set of elements \(x\in G\) such that \(\lim_{n\to\infty}[x,\underbrace{g,\ldots,g}_{n}]=1\) for every \(g\in G\). Moreover, a closed, core-free subgroup of \(G\) is shown to be contained in \(Z^\infty(G)\) if and only if it permutes with each Sylow subgroup and each Sylow normalizer.
scientific article; zbMATH DE number 6362522