Group actions
Group actions are vital to understanding and constructing quotient groups.
A group action of a group (G,\star) on a set A is a map G \times A \to A (denoted g \cdot a) such that for all g_1, g_2 \in G and a \in A
(1) g_1 \cdot (g_2 \cdot a) = (g_1 \star g_2) \cdot a, and
(2) e_G \cdot a = a
Where e_G is the identity element of G.
It is common for g \cdot a to be abbreviated ga. This may be confusing at times, as (g_1 \star g_2) \cdot a above could be abbreviated as g_1g_2a, so discretion is required.
Let G be a group acting on the set A. For every g \in G we have a map defined
\sigma_g \colon A \to A
\sigma_g(a) = g \cdot a.
Given the above definition for \sigma_g,
(1) For each fixed g \in G, \sigma_g is a permutation of A
(2) The map G \to S_A defined by g \rightarrowtail \sigma_g is a homomorphism.
(1) We prove \sigma_g is a permutation of A by showing
\sigma_g \colon A \times A
Is a bijection. We prove this by showing there exists a two-sided inverse \sigma_{g^{-1}}.
\begin{align*}
\sigma_{g^{-1}}(\sigma_g(a)) &= g^{-1}\cdot(g\cdot a)\\
&= (g^{-1}\star g) \cdot a\\
&= e_G \cdot a\\
&= a
\end{align*}
Since g was arbitrary, we may replace g with g^{-1} to obtain another identity map on A. Thus \sigma_g is a bijection, and a permutation.
(2) To show the map a homomorphism, we first define \varphi \colon G \to S_A, such that
\varphi(g) = \sigma_g.
Then through a series of equivalances
\begin{align*}
\varphi(g_1 \star g_2)(a) &= \sigma_{g_1g_2}(a)\\
&= (g_1 \star g_2) \cdot a\\
&= g_1 \cdot (g_2 \cdot a)\\
&= \sigma_{g_1}(\sigma_{g_2}(a))\\
&= (\varphi(g_1) \circ \varphi(g_2))(a)
\end{align*}
Thus the map is a homomorphism.
A group action of G on a set A intuitvly means that every element of G acts as a permutation on A, in some manner consistent with the group operation of G. The homomorphism from G to S_A given above is called the permutation representation associated to the given group action.
The definition given above could be more precisely called the left action as the group elements are left to the set elements. A right action could be defined identically.