Nilpotent and abelian Hall subgroups in finite groups. (Q2787978)

There are not many theorems analyzing local and global structure of a finite group from the point of view of two different primes.NEWLINENEWLINE Let \(G\) be a finite group, \(p\) and \(q\) be different primes and \(\pi\) be a set of primes. The main result of this paper is the following Theorem A: Some Sylow \(p\)-subgroup of \(G\) commutes with some Sylow \(q\)-subgroup of \(G\) if and only if the class sizes of the \(q\)-elements of \(G\) are not divisible by \(p\) and the class sizes of the \(p\)-elements of \(G\) are not divisible by \(q\).NEWLINENEWLINE Theorem A gives a characterization of the finite groups having nilpotent or abelian Hall \(\pi\)-subgroups that can easily be verified using the character table. This characterization can be seen as a contribution to \textit{Richard Brauer}'s Problem 11 from his celebrated paper [Lect. Modern Math. 1, 133-175 (1963; Zbl 0124.26504)]. In the course of proving Theorem A the authors show that for finite simple groups, commuting Sylow subgroups for different primes are actually always abelian.NEWLINENEWLINE Using Theorem A and the results of \textit{A. Moretó} [J. Algebra 379, 80-84 (2013; Zbl 1285.20019)], the authors deduce the following Theorem B: \(G\) has nilpotent Hall \(\pi\)-subgroups if and only if for every pair of distinct primes \(p,q\in\pi\), the class sizes of the \(p\)-elements of \(G\) are not divisible by \(q\). -- Theorem B gives an algorithm to determine from the character table if a group has nilpotent Hall \(\pi\)-subgroups.NEWLINENEWLINE As a consequence of Theorem B and the main results of \textit{G. Navarro} and \textit{P. H. Tiep} [J. Algebra 398, 519-526 (2014; Zbl 1305.20023)] and \textit{G. Navarro, R. Solomon} and \textit{P. H. Tiep} [J. Algebra 421, 3-11 (2015; Zbl 1308.20017)], it is obtained the following explicit way to detect from the character table of a finite group whether the group possesses an abelian Hall-subgroup (Theorem C): \(G\) has abelian Hall \(\pi\)-subgroups if and only if the two following conditions hold: (i) For every \(p\in\pi\) and every \(p\)-element \(x\in G\), \(|G:C_G(x)|\) is a \(\pi'\)-integer; (ii) If \(\{p\}=\pi\cap\{3,5\}\), then for every irreducible character \(\chi\) in the principal \(p\)-block of \(G\), \(\chi(1)\) is not divisible by \(p\).