Odd order Hall subgroups of the classical linear groups
Let \(G\) be one of the finite classical linear groups, i.e., of the full linear groups, the symplectic groups, the orthogonal groups, and the unitary groups. It is proved that \(G\) has a Hall \(\pi\)-subgroup \((\pi\subseteq\pi(G)-\{2\})\) if, and only if, \(G\) has a Hall \(\{r,s\}\)-subgroup for each pair of distinct primes \(r\) and \(s\) in \(\pi\). A necessary and sufficient condition for a classical linear group to have a Hall \(\{r,s\}\)-subgroup, where \(r\) and \(s\) are distinct primes, dividing the order of \(G\), and \(2\notin\pi(G)\), is also obtained.
scientific article; zbMATH DE number 856568
scientific article