The classification of p-local finite groups over the extraspecial group of order p3 and exponent p

All 𝑝-local finite groups of rank two for odd prime 𝑝

“…For r = 2, then S ∼ = Z/p × Z/p, which is resistant by Corollary 3.4. If r = 3, then S ∼ = p 1+2 + , and this case has been studied in [36]. For r ≥ 4 all the p-rank two maximal nilpotency class groups appear only at p = 3, and we use the description and properties given in Appendix A.…”

“…For P = γ 1 or γ 2 it is a monomorphism if P is F-Alperin as Out F (P ) is 3-reduced. For P = E i , V i they are inclusions by [36,Lemma 3.1] and by definition, respectively. In the case that this restriction is a monomorphism we identify Out F (P ) with its image in GL 2 (3) without explicit mention.…”

“…So every finite group gives rise to a p-local finite group, although there exist exotic p-local finite groups which are not associated to any finite group, as can be read in [11,Sect. 9], [30], [36] or Theorem 5.10 below. Besides its own interest, the systematic study of possible p-local finite groups, i.e.…”

Endotrivial modules for the sporadic simple groups and their covers

“…The outer automorphism group of P is isomorphic to GL 2 (p) and the matrix r ′ r s ′ s with determinant d corresponds to the class of automorphisms sending x to x r ′ y s ′ , y to x r y s and z to z d . Ruiz and Viruel [RV04] have classified the saturated fusion systems on P providing us with the information we need in order to compute n G . That is, the number of Gconjugacy classes of the p + 1 elementary abelian p-subgroups of rank 2.…”

The Dade group of a fusion system

“…Let P be an extraspecial group of order 7 3 and exponent 7. By [22], there is an exotic fusion system F on P, where the F-centric F-radical subgroups of P are the eight elementary abelian subgroups of order 49. In the same notation as above, with p ¼ 7, we may assume that E 1 and E 2 are F-conjugate, and E 3 ; .…”