8 BOB OLIVER Proposition 1.6 ([AKO, Theorem I.3.5]). Let F be a saturated fusion system over a p-group S. Then each morphism in F is a composite of restrictions of automorphisms in AutF(S), and in Op (AutF(P )) for P EF. Lemma 1.7. Let F be a saturated fusion system over a p-group S, and assume P EF. Then Op(OutF(P )) = 1, and OutF(P ) acts faithfully on P/Fr(P ). Proof. Since OutF(P ) has a strongly p-embedded subgroup, Op(OutF(P )) = 1 (cf. [AKO, Proposition A.7(c)]). The kernel of the action of AutF(P ) on P/Fr(P ) is a p-group by Lemma A.9, so OutF(P ) acts faithfully since Op(AutF(P )) = Inn(P ). The next two results give some necessary conditions for a subgroup to be es- sential. They were in fact proven in [OV] as conditions for a subgroup to be “critical”, but by [OV, Proposition 3.2], a subgroup of S which is F-essential for some saturated fusion system over S is a critical subgroup of S. Lemma 1.8 ([OV, Lemma 3.4]). Let F be a saturated fusion system over a p-group S. Let P S, let Θ be a characteristic subgroup in P , and assume there is g NS(P ) P such that (i) [g, P ] Θ·Fr(P ), and (ii) [g, Θ] Fr(P ). Then cg Op(Aut(P )), and hence P / EF. The proof of Lemma 1.8 is based on the fact that Op(OutF(P )) = 1 (Lemma 1.7). The next proposition is based on Bender’s classification [Be, Satz 1] of groups with strongly 2-embedded subgroups. Proposition 1.9 ([OV, Proposition 3.3(c)]). Let F be a saturated fusion sys- tem over a 2-group S. Fix P EF, and let k be such that |NS(P )/P | = 2k. Then rk(P/Fr(P )) 2k. 1.2. Reduced fusion systems We now consider the class of reduced fusion systems, as defined in [AOV1]. First recall the following definitions from [BCGLO2]. Definition 1.10. Let F be a saturated fusion system over a p-group S. (a) The focal subgroup of F is the subgroup foc(F) def = s−1t s, t S, t sF = s−1α(s) s P S, α AutF(P ) . (b) The hyperfocal subgroup of F is the subgroup hyp(F) = s−1α(s) s P S, α Op(AutF(P )) . For any saturated fusion subsystem F0 F over a subgroup S0 S, (c) F0 has p-power index in F if S0 hyp(F), and AutF 0 (P ) Op(AutF(P )) for all P S0 and (d) F0 has index prime to p in F if S0 = S, and AutF 0 (P ) Op (AutF(P )) for all P S.
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