SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

(b) Suppose E is a normal subgroup of G such that p divides |E|. Then E �≤ H.
Also, since P < NT (P ) for every proper subgroup P of T , 2.7.2(a) implies
Remark 2.7.3. If H is strongly p-embedded in G and T ∩ H �= 1, then T ≤ H. In
particular, every strongly p-embedded subgroup of G contains a Sylow p-subgroup of
G.
Together with 2.7.2 this implies
Remark 2.7.4. If G has a strongly p-embedded subgroup, then Op(G) = 1.
In view on 2.7.2(b), the normalizers of non-trivial p-subgroups of H play an im-
portant role in the understanding of strongly p-embedded subgroups. We define now
R(S, G) := 〈NG(P ) : 1 �= P ≤ S〉,
for every S ∈ Sylp(G). This leads to a new characterization of strongly p-embedded
subgroups, as stated in the following lemma.
Lemma 2.7.5. Assume H ∩ T �= 1. Then H is strongly p-embedded if and only if H is
a proper subgroup of G and R(T, G) ≤ H.
Proof. See [14, 17.11].
Corollary 2.7.6. The group G contains a strongly p-embedded subgroup if and only if
R(T, G) �= G.
Proof. If H is a strongly p-embedded subgroup of G, then by 2.7.3, T g ≤ H for
some g ∈ G. Thus, by 2.7.5, R(T g , G) �= G. Since R(T g , G) = R(T, G) g , it follows
R(T, G) �= G. On the other hand, if R(T, G) �= G, then by 2.7.5, R(T, G) is a strongly
p-embedded subgroup of G.
Example 2.7.7. Assume G ∼ = SL2(q) and H = NG(T ). Then H is a strongly p-
embedded subgroup of G.
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