SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

Lemma 2.7.10. Let T ≤ H ≤ G be strongly p-embedded and N ✂ G such that
HN �= G. Then N is a p ′ group and HN/N is strongly p-embedded in G/N.
Proof. We have T ∩N ∈ Sylp(N) and by the Frattini Argument, G = NNG(T ∩N) �=
HN. Hence, NG(T ∩ N) �≤ H and therefore, since H is strongly p-embedded in G,
T ∩ N = 1. Hence, N is a p ′ -group.
We now set G = G/N. Let 1 �= P ≤ T . We need to show that N G (P ) ≤ H.
Let g ∈ G such that g ∈ N G (P ). Then P g N = (P N) g = P N. Since N is a p ′ -
group, P ∈ Sylp(NP ). Hence, there is h ∈ NP such that P gh = P . Then gh ∈ H,
since H is strongly p-embedded. Moreover, h −1 ∈ P N and P ≤ T ≤ H. Hence,
g = (gh)h −1 ∈ HN and g ∈ H.
Lemma 2.7.11. Assume G has a strongly p-embedded subgroup. Let N ✂ G be such
that N = E1 × . . . × Er for subgroups E1, . . . , Er of G. If p | |Ei| for i = 1, . . . , r,
then r = 1.
Proof. Let H ≤ G be strongly p-embedded such that T ≤ H. Since p divides |N|,
N �≤ H. Hence, there is i ∈ {1, . . . , r} such that Ei �≤ H. Assume r > 1. Then there
is i �= j ∈ {1, . . . , r}. Since Ej ✂ N and N ✂ G, Ej is subnormal in G. Therefore,
T ∩ Ej ∈ Sylp(G). In particular, T ∩ Ej �= 1. Since [Ei, Ej] = 1, it follows that
Ei ≤ NG(Ej ∩ T ) ≤ H, a contradiction.
Lemma 2.7.12. Let H ≤ G be strongly p-embedded in G and assume T ≤ H. Sup-
pose E is a normal subgroup of G such that E ∼ = SL2(q). Then H = NG(T ∩ E) and
G/E is a p ′ -group.
Proof. Set T0 = E ∩ T . The Frattini Argument gives us G = ENG(T0). By 2.7.2(a)
and 2.7.3, NG(T0) ≤ H. So it follows from Dedekind’s Law that H = NG(T0)(E∩H).
Assume E ∩ H �≤ NG(T0). Then there is T0 �= T1 ∈ Sylp(E ∩ H). Hence, 2.6.3 yields
E = 〈T0, T1〉 ≤ H. This is a contradiction since q divides |E|, E is normal in G and
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