Classification and lattice properties of pronormal subgroups in PSL(2,q), J1, and Sz(q) for the specified values of q
Yuto Nogata
TL;DR
The paper delivers a comprehensive classification of pronormal subgroups in PSL$(2,q)$, Sz$(q)$, and J$_1$ for specified $q$-ranges, showing that while the pronormal family is closed under joins, it typically fails to be closed under meets; in each case a lattice can be obtained by adopting a suitable meet operation. Across PSL$(2,q)$, the pronormal structure distinguishes between PH and NPr regimes, with many odd-order cyclic and large dihedral subgroups pronormal, but certain elementary abelian $p$-subgroups and 2-subgroups obstruct pronormality; the meet of two pronormal subgroups can yield a non-pronormal subgroup. In J$_1$, all subgroups are pronormal except $Z_2$ and $(Z_2)^2$, and the join of pronormal subgroups remains pronormal while the meet can fail. For Sz$(q)$ with $q=2^{2n+1}$ and $2n+1$ prime, most subgroups are pronormal except some 2-subgroups; the non-nilpotent pronormal subgroups form a tractable class whose join is pronormal, and a family-wide lattice is achieved via a canonical meet operation. Overall, these results refine our understanding of embedding properties in key finite simple groups and offer a concrete pathway to lattice-like structures through adjusted meet notions.
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.