Group Actions, An Introduction
Introduction
A group G acts on a set S if each element in G is associated
with a permutation on the elements of S.
The group identity is mapped to the identity permutation,
and the group element a*b
drives the permutation of a followed by the permutation of b.
Since b/b = 1, the permutation of b inverse is the inverse of the permutation of b.
In fact the image of G is a permutation group,
and group action commutes with * by definition,
so the action defines
a group homomorphism from G into the symmetric group on S.
The group elements that map to the identity permutation form the kernel.
We say G acts effectively on S if the homomorphism is 1-1;
every member of G (other than 1) induces a nontrivial permutation on S.
This is also written "G embeds in S",
which really means G embeds in the permutation group of S.
One can reverse the sense of group action, so that the action of a*b is the action of b followed by the action of a.
Note that G still maps (homomorphically) into the permutation group on S,
provided the permutation group is defined in such a way that
the composition of two permutations applies the second, and then the first.
Oddly enough, this backwards convention is used more often than the forward convention.
It is convenient in certain applications, as we shall see later on.
Let G act on S in the forward direction,
and build a new group action that runs in reverse.
First, invert each permutation in the action of G.
Then, let a*b apply the permutation of b followed by the permutation of a.
Now G acts on S using the reverse convention.
Each forward action implies a reverse action, and vice versa.
It is merely a matter of taste.