On Oliver's \(p\)-group conjecture for Sylow subgroups of symplectic groups and orthogonal groups (Q6612144)

The following conjecture, now known as Oliver's \(p\)-group conjecture, was proposed by \textit{B. Oliver} [Math. Proc. Camb. Philos. Soc. 137, No. 2, 321--347 (2004; Zbl 1077.55006), Conjecture 3.9]: Let \(p \not =2\) be a prime and \(S\) be a \(p\)-group. Then \(J(S)\leq \mathfrak{X}(S)\), where \(J(S)\) is the Thompson subgroup and \(\mathfrak{X}(S)\) is the Oliver subgroup described in [loc. cit.].\N\N\textit{D. J. Green} et al. [J. Algebra 342, No. 1, 1--15 (2011; Zbl 1258.20014)] showed that Oliver's conjecture [loc. cit.] holds for the Sylow subgroups of the symmetric groups and the general linear groups.\N\NThe main result of the paper under review is that if \(m \geq 1\), \(p \geq 5\) is a prime, \(q\) is any prime power and \(S\) is a Sylow \(p\)-subgroup of \(\mathrm{Sp}_{2m}(\mathbb{F}_{q})\), \(\mathrm{O}_{2m}^{\pm}(\mathbb{F}_{q})\) or \(\mathrm{O}_{2m+1}(\mathbb{F}_{q})\), then Oliver's conjecture [loc. cit.] holds for \(S\).