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

TL;DR

This paper investigates the pronormality of subgroups of odd index in finite simple symplectic groups, establishing conditions under which such subgroups are pronormal or nonpronormal based on the group's parameters.

Contribution

It extends previous work by characterizing when subgroups of odd index are pronormal in symplectic groups, identifying specific cases with nonpronormal subgroups.

Findings

Subgroups of odd index are pronormal in PSp_{2n}(q) when n is a power of two.

Nonpronormal subgroups of odd index exist in PSp_{2n}(q) for certain n not of the specified forms.

The pronormality depends on the structure of n and the congruence class of q modulo 8.

Abstract

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g\rangle$ 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 $PSp_{6n}(q)$ with $q \equiv \pm 3 \pmod 8$ contains a nonpronormal subgroup of odd index. In this paper we prove that if $n \not \in \{2^m, 2^m(2^{2k}+1) \mid m, k \in \mathbb{N} \cup \{0\}\}$ then a group $PSp_{2n}(q)$ with $q \equiv \pm 3 \pmod 8$ contains a nonpronormal subgroup of odd index; if $n=2^m$ then any subgroup of odd index is pronormal in a group $PSp_{2n}(q)$, where $q \equiv \pm 3 \pmod 8$.