Section 8.1
Permutation groups
The purpose of this section is
to show that every finite group
can be viewed as a group of permutations.
Let X be a set.
Recall that by Sym(X) we denote the group of all
bijections from X to X itself.
Definition
- A permutation group on X is a subgroup of Sym(X).
- If H is a group and f: H -> Sym(X) a morphism of groups, then we call f a permutation representation of H (on X).
In both cases the group is said to
act on X. We also speak of the `action' of the group on
X. The elements of the set X are called
points.
The size of X, denoted by |X|, is called the degree of the representation.
If f is a permutation representation
of the group G on
X, and g 
G, then we often
write g(x), or just g x, for the image of
x under g, instead of the more complete
expression f(g) (x).