Abelian Sylow subgroups in a finite group. (Q401755)

The authors prove the following theorem: If \(p\neq 3\), \(5\) is a prime then Sylow \(p\)-subgroups of a finite group \(G\) are Abelian if and only if the class sizes of the \(p\)-elements of \(G\) are all coprime to \(p\). This gives a solution (for \(p\neq 3\), \(5\)) to a problem posed by \textit{R. Brauer} in 1956 (see Problem 12 in [Lect. Modern Math. 1, 133-175 (1963; Zbl 0124.26504)]). The theorem was obtained by \textit{A. R. Camina} and \textit{M. Herzog} [Proc. Am. Math. Soc. 80, 533-535 (1980; Zbl 0447.20004)] for \(p=2\).NEWLINENEWLINE The sporadic simple groups \(G=J_4\) for \(p=3\) and \(G=Th\) for \(p=5\) do not have Abelian Sylow \(p\)-subgroups, yet the centralizers of the \(p\)-elements do contain full Sylow \(p\)-subgroups. Thus the theorem does not hold for \(p\in\{3,5\}\).