SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

Chapter 2
Preliminaries on Groups
Throughout this thesis we write function on the right side. In this chapter, if φ is a
homomorphism from a group G, we usually write g φ rather than gφ for the image of g
under φ. Similarly, we write G φ for the image of G under φ.
The aim of this chapter is to give an overview on some basic group theoretical
background and to fix some notation as necessary. In that we follow [15]. We also
prove some more specialized results which we will need later on. In particular, in
Section 2.9 we show the uniqueness of certain amalgams.
In the remainder of this chapter G will always be a finite group, p a prime and T a
Sylow p-subgroups of G. Moreover, q is always assumed to be a power of p.
2.1 Notation and basic results
We will write o(g) for the order of an element g ∈ G, and Sylp(G) for the set of Sylow
p-subgroups of G. The group G is called p-closed if T is normal in G. We will make
use of the following characteristic subgroups of G:
• Op(G) is the largest normal p-subgroup of G.
• O p (G) is the smallest normal subgroup of G whose factor group is a p-group.
Equivalently, O p (G) is the group generated by all p ′ -elements of G.
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