SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

The following lemma shows that, given a group G which is not p-closed, it is easy
to obtain minimal parabolic subgroups of G containing a Sylow p-subgroup of G.
Lemma 2.8.3. Let G0 be a subgroup of G such that NG(T ) ≤ G0 < G. Assume that
P is a subgroup of G which is minimal with the properties T ≤ P and P �≤ G0. Then
P is minimal parabolic and G0 ∩ P is the unique maximal subgroup of P containing
T .
Proof. We have T ∈ Sylp(P ). Since P �≤ G0 and NG(T ) ≤ G0, T is not normal in
P . Now let M be a maximal subgroup of P containing T , and assume M �= G0 ∩ P .
Then, since M is maximal, M �≤ G0 ∩ P . Hence, M �≤ G0. Also, T ≤ M < P , a
contradiction to the minimality of P .
2.9 Uniqueness of amalgams
Throughout this section, A = (G1, G2, G12, φ1, φ2) and B = ( ˜ G1, ˜ G2, ˜ G12, ψ1, ψ2) are
amalgams, as defined in the introduction.
Definition 2.9.1. A triple (β1, β2, β) of group isomorphisms β : G12 → ˜
G12 and
βi : Gi → ˜ Gi, for i = 1, 2, is said to be an isomorphism from A to B, if the obvious
diagram commutes, i.e. if φiβi = βψi, for i = 1, 2.
Recall that G1 ∗G12 G2 denotes the free product of G1 and G2 with G12 amal-
gamated. Here we suppress mention of φ1, φ2 and thus of the exact isomorphism
type of A. However, the amalgam should always be clear from the context. Usually
we identify G1, G2 and G12 with their images in G1 ∗G12 G2, so the monomorphisms
φi : G12 → Gi become inclusion maps which we, by abuse of notation, denote by id.
Also, if H is a group and φ : G12 → H is a group isomorphism, then A is isomorphic
to (G1, G2, H, φ −1 φ1, φ −1 φ2) via (id, id, φ). Thus, if we want to show that A and B
are isomorphic, we usually may assume that G12 = ˜ G12, and that G12 is contained in
G1, G2, ˜ G1 and ˜ G2. The next remark deals with this situation.
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