Kernel

The kernel of the action is the set of elements of \(G\) that act trivially on every element of \(A\):\(\{g\in G \mid g\cdot a = a \text{ for all } a \in A\}\).

Stabilizer

For each \(a \in A\), the stabilizer of \(a\) in \(G\) is the set of elements of \(G\) that fix the element \(a\): \(\{g \in G \mid g \cdot a = a\}\) and is denoted by \(G_a\).

Faithful

An action is faithful if its kernel is the identity.

Proposition 1

For any group \(G\) and any nonempty set \(A\), there is a bijection between the actions of \(G\) on \(A\) and the homomorphisms of \(G\) into \(S_A\).

Proof

Earlier, we showed that,

• for each \(g \in G\), \(\sigma_g: A \rightarrow A\) defined by \(\sigma_g(a) = g\cdot a\) is an element of \(S_A\).
• the map from \(G\) to \(S_A\) defined by \(g \mapsto \sigma_g\) is a homomorphism.

(See Group action)

This establishes the one-to-one mapping between the actions of \(G\) on \(A\) and the homomorphism \(g \mapsto \sigma_g\). In other words, if \(\varphi\) is the homomorphism \(g \mapsto \sigma_g\), we can get an action of \(G\) on \(A\) by $$ g \cdot a = \varphi(g)(a). $$

$$\tag*{$\blacksquare$}$$

Permutation representation

If \(G\) is a group, a permutation representation of \(G\) is any homomorphism of \(G\) into the symmetric group \(S_A\) for some nonempty set \(A\). We say a given action of \(G\) on \(A\) affords or induces the associated permutation representation of \(G\).

Proposition 2

Let \(G\) be a group acting on the nonempty set \(A\). The relation on \(A\) defined by $$ a \sim b\;\text{ if and only if }\; a = g \cdot b\;\text{ for some }\; g \in G $$ is an equivalence relation. For each \(a \in A\), the number of elements in the equivalence class containing \(a\) is \(|G:G_a|\), the index of the stabilizer of \(a\).

Proof

For \(a \in A\), \(1 \in G\) and \(a = 1 \cdot a\). Therefore, \(\sim\) is reflexive.

For any \(a,b \in A\),

$$ a \sim b \implies a = g \cdot b \implies g^{-1} \cdot a = g^{-1} \cdot (g \cdot b) = (g^{-1}g) \cdot b = b \implies b \sim a $$

for some \(g \in G\). Therefore, \(\sim\) is symmetric.

For any \(a,b,c \in A\), if \(a \sim b\) and \(b \sim c\), then \(a = g_1 \cdot b\) and \(b = g_2 \cdot c\) for some \(g_1, g_2 \in G\). It follows that \(a = g_1 \cdot b = g_1 \cdot (g_2 \cdot c) = (g_1g_2) \cdot c\), and thus \(a \sim c\). Therefore, \(\sim\) is transitive.

Hence, \(\sim\) is an equivalence relation. $$\tag*{$\blacksquare$}$$

Let \(C_a\) be the equivalence class containing \(a \in A\). That is, \(C_a = \{g\cdot a\ \mid g \in G\}\).

To show that the last assertion is true, it suffices to show that there exists a bijection between \(C_a\) and the set of left cosets of \(G_a\).

Let \(\varphi\) be the mapping from \(C_a\) to the set of left cosets of \(G_a\) defined by \(\varphi(b) = gG_a\), where \(g\) is an element of \(G\) such that \(b = g \cdot a\).

\(\varphi\) is surjective because for any \(g \in G\), \(g \cdot a \in C_a\).

For any \(g,h \in G\), \(g \cdot a = h \cdot a \iff h^{-1}g \cdot a = a \iff h^{-1}g \in G_a\). By Proposition 3.4, \(h^{-1}g \in G_a \iff gG_a = hG_a\). Therefore, \(\varphi\) is injective.

Hence, \(\varphi\) is bijective.

$$\tag*{$\blacksquare$}$$

Orbit

Let \(G\) be a group acting on the nonempty set \(A\). The equivalence class \(\{g \cdot a \mid g \in G\}\) is called the orbit of \(G\) containing \(a\).

Transitive action

Let \(G\) be a group acting on the nonempty set \(A\). The action of \(G\) on \(A\) is called transitive if there is only one orbit, i.e., given any two elements \(a,b \in A\), there is some \(g\in G\) such that \(a = g \cdot b\).