The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle
Abstract
Let G be an almost simple group whose socle is an alternating, classical, or sporadic group, and let H be a non-parabolic maximal subgroup of G. We prove that any maximal subgroup M of H can be generated by at most $7$ elements, and that this bound is sharp when the socle of G is alternating or classical; this improves the bound of $12$ due to Burness, Liebeck and Shalev. When the socle is sporadic, at most $5$ generators suffice, and this is again best possible. The proof relies upon a detailed structural analysis of H and M, especially when H is a subgroup of a wreath product. In particular, we determine the chief factors of M and subsequently bound its number of generators using the theory of crowns. We also correct the classification of maximal subgroups H of almost simple groups requiring more than three generators, established by Lucchini, Marion and Tracey. Their bound of five generators remains valid, but their classification is missing several pairs (G,H), including cases in which G has socle PSUn(q).
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BibTeX
@misc{capilla2026structure,
title = {{The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle}},
author = {Patricia Medina Capilla},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.19105},
eprint = {2608.19105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19105},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle
AU - Patricia Medina Capilla
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.19105
UR - https://arxiv.org/abs/2608.19105
AB - Let G be an almost simple group whose socle is an alternating, classical, or sporadic group, and let H be a non-parabolic maximal subgroup of G. We prove that any maximal subgroup M of H can be generated by at most $7$ elements, and that this bound is sharp when the socle of G is alternating or classical; this improves the bound of $12$ due to Burness, Liebeck and Shalev. When the socle is sporadic, at most $5$ generators suffice, and this is again best possible. The proof relies upon a detailed structural analysis of H and M, especially when H is a subgroup of a wreath product. In particular, we determine the chief factors of M and subsequently bound its number of generators using the theory of crowns. We also correct the classification of maximal subgroups H of almost simple groups requiring more than three generators, established by Lucchini, Marion and Tracey. Their bound of five generators remains valid, but their classification is missing several pairs (G,H), including cases in which G has socle PSU_n(q).
ER -