On the pronormality of subgroups of odd indices in finite simple symplectic groups

Abstract

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $⟨H,H^g⟩$ for every element $g\in G$. In [Sib. Math. J. 2015. Vol. 56, no. 6] we proved that subgroups of odd indeces are pronormal in many finite simple group. In [Proc. Steklov Inst. Math., to appear] we proved that a group $PS{p}_{6n}(q)$ with $q\equiv \pm 3(\mathrm{mod}8)$ contains a nonpronormal subgroup of odd index. In this paper we prove that if $n\notin \{2^m,2^m(2^{2k}+1)\mid m,k\in \mathbb{N}\cup \{0\}\}$ then a group $PS{p}_{2n}(q)$ with $q\equiv \pm 3(\mathrm{mod}8)$ contains a nonpronormal subgroup of odd index; if $n=2^m$ then any subgroup of odd index is pronormal in a group $PS{p}_{2n}(q)$, where $q\equiv \pm 3(\mathrm{mod}8)$.

14 pages, in Russian

References in corpus (1)

• Pronormality of Hall subgroups in finite simple groups

Cited by in corpus (1)

• Finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal