Sylow numbers of finite groups
For a finite group \(G\) and a prime \(p\) dividing the order of \(G\), the Sylow \(p\)-number of \(G\) is \(n_p=|G:N_G(P)|\), where \(P\) is any Sylow \(p\)-subgroup of \(G\). The study of these numbers was initiated by \textit{Ph. Hall} [J. Lond. Math. Soc. 3, 98-105 (1928; JFM 54.0145.01)] and continued by \textit{M. Hall jun.} [J. Algebra 7, 363-371 (1967; Zbl 0178.02102)]. The elegant paper under review is the first post-classification attempt to revisit the Sylow numbers; induction and the classification of the simple groups are used to prove in Section 2 the following beautiful arithmetic criterion for \(p\)-nilpotency: A finite group \(G\) is \(p\)-nilpotent for some prime \(p\) if and only if \((p,n_q)=1\) for all prime divisors \(q\) of the order of \(G\). The finite groups in which the Sylow numbers are square-free and those in which the Sylow numbers are prime powers are classified in Sections 4 and 3, respectively. As a consequence, it follows for example that a solvable group with square-free Sylow numbers must be supersolvable. In the last section, the nilpotent length, the derived length and the \(p\)-length of a finite solvable group are bounded in terms of certain invariants related to the Sylow numbers of the group.
scientific article; zbMATH DE number 802371