The AWC-goodness and essential rank of sporadic simple groups

摘要

In a recent paper, Navarro and Tiep defined the property AWC-good for finite simple groups. They proved that the Alperin Weight Conjecture holds for every finite group if every finite simple group is AWC-good. We show that every sporadic simple group is AWC-good. Our computational proof requires to construct many radical subgroups of sporadic simple groups up to conjugacy; we provide these groups in an extensive appendix. As another application, for every sporadic simple group G and prime p, we determine the essential p-rank of G, that is, the number of G-conjugacy classes of essential subgroups of a Sylow p-subgroup D of G. The essential p-rank is closely related to the minimal cardinality of a conjugation family for the Frobenius category FG(D).