Classification and lattice properties of pronormal subgroups in PSL(2,q), J1, and Sz(q) for the specified values of q

TL;DR

This paper completes the classification of pronormal subgroups in PSL(2,q), Sz(q), and J1 groups, revealing their lattice properties and how they behave under joins and meets.

Contribution

It provides a complete classification of pronormal subgroups in these groups and analyzes their lattice structure and closure properties.

Findings

Pronormal subgroups are closed under joins.

They are not closed under meets.

Replacing meet with a different operation forms a lattice.

Abstract

We complete the classification of pronormal subgroups in the projective special linear groups PSL(2,q), the Suzuki groups of Lie type Sz(q), and the first Janko group J1, for the same ranges of q as in previous studies. Building on those works, we settle the remaining cases under the same parameter conditions. For each of these finite simple groups, the family of pronormal subgroups is closed under joins but not under meets. If the meet operation is replaced by a suitable operation, the family becomes a lattice.