Locally finite simple groups whose nonnilpotent subgroups are pronormal (Q6564344)

This paper investigates the classification of locally finite simple groups where the non-nilpotent subgroups are either pronormal or nilpotent. A \textit{pronormal subgroup} \( H \) of a group \( G \) is defined as a subgroup for which every conjugate \( H^g \) of \( H \) within \( G \) is conjugate to \( H \) in the subgroup generated by \( H \) and \( H^g \), that is, \( H \) is conjugate to \( H^g \) in \( \langle H, H^g \rangle \). Pronormal subgroups include normal and maximal subgroups and all Sylow subgroups of finite groups are pronormal.\N\NThe main results of the paper are the following.\N\NTheorem 1.1. Let \( G \) be a non-abelian finite simple group. Then \( G \) has only pronormal or nilpotent subgroups if and only if it is isomorphic to one of the following groups:\N\begin{itemize}\N\item \(\mathrm{PSL}(2, q) \) where \( q \) satisfies:\N\begin{itemize}\N\item[(i)] \( q = 2^d \), \( d \) prime;\N\item[(ii)] \( q = 3^d \), \( d \) an odd prime;\N\item[(ii)] \( q \) prime, and \( q \equiv \pm 1 \mod 8 \), where either \( q - 1 \) or \( q + 1 \) is a power of 2.\N\end{itemize}\N\N\item The sporadic group \( J_1 \).\N\item Suzuki groups \(\mathrm{Sz}(q) \), where \( q = 2^{2n+1} \) and \( 2n + 1 \) is prime.\N\end{itemize}\N\NTheorem 1.2. No infinite locally finite simple group has only nilpotent or pronormal subgroups.\N\NThese results extend previous results about locally finite groups in which every subgroup is either pronormal or abelian.