Group Action
A group action of a group $ G $ on a set $ X $ is a mapping $ \circ: G \times X \to X $ such that the following axioms hold:
1. $ e \circ x = x ~~ \forall x \in X $ where $ e $ is the identity of $ G $ 2. $ (g_1 \cdot g_2) \circ x = g_1 \circ (g_2 \circ x) ~~ \forall x\in X, g_1,g_2\in G $
In many cases, there is a very natural and intuitive action for a group. For example, when $ G = S_n $, a symmetric group, since the elements are themselves functions on $ \{1, 2, \ldots, n\} $ (or an arbitrary n element set), it is natural to take $ X = \{1, 2, \ldots, n\} $ and define the action as evaluation of the permutation. Since the operation of $ G $ is function composition and the identity element maps every element to itself, it is clear that the group action axioms hold. As an example, let $ g = (1 2 3) \in S_3, x = 2 $. Then $ g \circ x = 3 $ since $ (1 2 3) $ takes $ 2 \mapsto 3 $.
--Jvaught 21:09, 30 March 2010 (UTC)
