SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

By 2.5.2, every non-trivial element of P GL2(q) induces a non-trivial automor-
phism of SL2(q) and P SL2(q), i.e. P GL2(q) embeds into the automorphism group of
SL2(q) and P SL2(q).
Also note that Aut(F ) acts on GL2(q), SL2(q), P GL2(q) and P SL2(q). The ac-
tion of Aut(F ) on these groups is faithful. In particular, we can form the semidirect
product Aut(F ) ⋉ P GL2(q) and it embeds into Aut(SL2(q)).
Similarly, we can find an embedding of Aut(F ) ⋉ P GL2(q) into Aut(P SL2(q)).
By a classical result we have equality as described in the following theorem.
Theorem 2.6.7. Aut(SL2(q)) ∼ = Aut(P SL2(q)) ∼ = Aut(F ) ⋉ P GL2(q).
We remind the reader that for q = p n , Aut(F ) is cyclic of order n and generated
by the so called Frobenius automorphism of F which takes every element of F to its
p-th power.
The next three lemmas are also well known results and we state them without a
proof or reference.
Lemma 2.6.8. Let q be odd. Then P GL2(q) and P SL2(q) have dihedral Sylow 2-
subgroups.
Lemma 2.6.9. Let q be odd. Then there is a unique extension of P SL2(q 2 ) of degree
2 with semidihedral Sylow 2-subgroups.
Lemma 2.6.10. Let p �= 2. Then all involutions in L(V ) are conjugate in L(V ).
on.
We conclude this section with a rather specialized result which we will need later
Lemma 2.6.11. Let E be a normal subgroup of G such that G = ET and E ∼ = SL2(q)
for some power q of p. Assume that NG(T ∩ E) is p-closed. Then G = EOp(G).
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