Odd order Hall subgroups of \(GL(n,q)\) and \(Sp(2n,q)\)
Consider the group \(GL(n,q)\), \(q=p^ f\), \(p\) a prime. Let \(\omega\) be a set of primes with \(2,p\not\in\omega\). The author shows that if \(GL(n,q)\) possesses \(\omega\)-Hall subgroups, then any \(\omega\)-subgroup of \(GL(n,q)\) has a normal Abelian \(\tau\)-Hall subgroup where \(\tau=\omega-\{r\}\) and \(r\) is the smallest prime of \(\omega\). This in turn is used to prove that all \(\omega\)-Hall subgroups are conjugate. Moreover the author shows that a stronger conjecture on Hall subgroups is not true: For any set \(\omega\) of odd primes there is an \(n\) and a \(q\) such that the pair \(GL(n,q),\omega\) satisfies the above conditions, but there are \(\omega\)-subgroups which are not even isomorphic to any subgroup of an \(\omega\)-Hall subgroup of \(GL(n,q)\). Similar results are proved for \(Sp(2n,q)\). By this paper some results of \textit{E. L. Spitznagel} [Math. Z. 97, 259-290 (1967; Zbl 0183.03104)] are generalized.
scientific article; zbMATH DE number 3873580
scientific article