SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

Chapter 3
Groups Acting on Modules
Throughout this chapter let G be a group, p a prime and T ∈ Sylp(G).Furthermore, let
F be a finite field of characteristic p and V a finite dimensional F G-module.
3.1 Preliminaries
Notation 3.1.1. Write End F (V ) for the set of F -linear transformations from V to
itself. Set
End F G(V ) := {φ ∈ End F (V ) : (vφ) g = (v g )φ for every g ∈ G}.
If this does not lead to any confusion we will write EndG(V ) instead of EndF G (V ).
If V is irreducible, i.e. if 0 and V are the only G-invariant F -subspaces of V , then
by Schur’s Lemma, EndF G (V ) is a skew field. Moreover, by Wedderburn’s Theorem,
every finite skew field is a field. Thus, if V is irreducible, K := EndF G (V ) is a finite
field and V is an KG-module.
Also note that multiplication by an element of F is an F -endomorphism of V which
commutes with every element of G. Therefore F is isomorphically contained in K and
can be regarded as a subfield of K.
Lemma 3.1.2. Let A and B be finite irreducible F G-modules and let φ : A → B be
an F G-module isomorphism. Set FA := EndF G (A) and FB := EndF G (B). Then φ is
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