SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

Lemma 2.2.3. For subgroups X, Y of G, [X, Y ] is a normal subgroup of 〈X, Y 〉.
Proof. See for example [15, 1.5.5].
Lemma 2.2.4 (Three-Subgroup Lemma). Let X, Y, Z ≤ G and let N be a normal
subgroups of G. Assume [X, Y, Z] ≤ N and [Y, Z, X] ≤ N. Then [Z, X, Y ] ≤ N.
Proof. This follows from [15, 1.5.6].
If a group A acts on the group G, then we can consider at A and G as subgroups of
the semidirect product A ⋉ G. Thus, the definitions of commutators and commutator
subgroups can be inherited and the results hold accordingly.
In particular, the Three-Subgroup Lemma holds, i.e. if each of X, Y, Z is a sub-
group of either G or A then
2.3 Coprime action
[X, Y, Z] = [Y, Z, X] = 1 =⇒ [Z, X, Y ] = 1.
In this section A will always be a group which acts on G. We call the action of A on
G coprime if
(1) |A| and |G| are coprime, and
(2) A or G is soluble.
We remind the reader here that a finite group G is called soluble if [U, U] < U for
all subgroups U of G. Thus abelian groups and nilpotent groups provide examples of
soluble groups. In particular, p-groups are soluble.
By a Theorem of Feit-Thompson, every finite group of odd order is soluble. Hence
if |K| and |G| are coprime then either K or G is soluble. Therefore assumption (2)
is in fact redundant. However we prefer to put it this way since the Theorem of Feit-
Thompson is a deep result of finite group theory and in all the situations we are going
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