We come to one of the central concepts in group theory, that of an action of a group on a set.
Definition5.1.Group Action.
For a group \(G\) and set \(S\text{,}\) an action of \(G\) on \(S\) is a function
\begin{equation*}
G \times S \to S,
\end{equation*}
typically written as \((g,s) \mapsto g \cdot s\text{,}\) such that
\(g \cdot (g' \cdot s) = (g g') \cdot s\) for all \(g,g' \in G\) and \(s\in S\text{.}\)
\(e_G \cdot s = s\) for all \(s \in S\text{.}\)
Example5.2.Group Actions.
Trivial Action.
For any group \(G\) and any set \(S\text{,}\)\(g \cdot s := s\) defines an action, the trivial action.
The group \(S_n\) acts on the set \(X=\{1,2,\dots,n\}\) such that \(\sigma\cdot i=\sigma(i)\text{.}\)
Left Multiplication.
For any group \(G\) and any set \(S\text{,}\)\(g \cdot s := gs\) defines an action, the left multiplication action. 1
And similarly for right multiplication, yet we do not speak of that here.
Conjugation.
For any group \(G\) and any set \(S\text{,}\)\(g \cdot s := gxg\inv\) defines an action, the conjugation action.
Answer.
Let \(G\) be a group and \(S\) a set.
Let \(g,g'\in G\) and \(s\in S\text{.}\) Notice \(g \cdot (g' \cdot s) = g\cdot s = s\) and \(e_G\cdot s = s\text{.}\) Thus the trivial action is indeed a group action.
This is a small assemblage with no title, simply to make sure the surrounding box behaves properly, especially for LaTeX output.
Remark5.3.
Though the set \(S\) need not be a group, it certainly can be. Indeed, a group \(G\) can act on itself, as we will see in Section 5.2.
SubsectionThe Permutation Representation
One can also think of group actions as homomorphisms.
Theorem5.4.Permutation Representation.
Assume \((G, \cdot)\) is a group and \(S\) is a set.
If \(\cdot\) is an action of \(G\) on \(S\text{,}\) then the function \(\rho: G \to \operatorname{Perm}(S)\) defined as \(\rho(g) = \sigma_g\text{,}\) where \(\sigma_g:S\to S\) is the function given by \(\sigma_g(s)=g \cdot s\text{,}\) is a well defined homomorphism of groups.
Conversely, if \(\rho: G \to \operatorname{Perm}(S)\) is a group homomorphism, the rule \(g \cdot s := \rho(g)(s)\) defines an action of \(G\) on \(S\text{.}\)
Proof.
We have
\begin{equation*}
\begin{aligned}
(\sigma_g\circ\sigma_{g^{-1}})(s) &=\sigma_g(\sigma_{g^{-1}}(s)) & \text{ (def of composition)}\\
&=g\cdot (g^{-1} \cdot s) & (\text{ def for } \sigma_g \text{ and } \sigma_{g^{-1}})\\
&=(gg^{-1})\cdot s & \text{ (first property of a group action)}\\
&=e_G\cdot s & \text{ (group axiom)}\\
&= s &\text{ (second property of a group action)}
\end{aligned}
\end{equation*}
thus \(\sigma_g\circ\sigma_{g^{-1}}=\text{id}_S\) and a similar argument shows that \(\sigma_{g^{-1}}\circ\sigma_{g}=\text{id}_S\)
Finally, we wish to show \(\rho(gg')=\rho(g) \circ \rho(g')\text{,}\) equivalently \(\sigma_{gg'}=\sigma_g\circ\sigma_{g'}\text{.}\) Since
holds for all \(s\text{,}\) this proves \(\rho\) is a homomorphism.
Given a homomorphism \(\rho\text{,}\) the function \(G \times S \to S\) defined as \(g \cdot s = \rho(g)(s)\) is an action because \(g'(gs) = \rho(g')(\rho(g)(s)) = (\rho(g') \circ\rho(g))(s) = \rho(gg')(s) = (gg')s\text{,}\) and \(e_G s = \rho(e_G)(s) = \text{id}(s) = s\text{.}\)
An immediate perk of this result is that we already know a lot of things about homomorphisms, and we can thus extend that knowledge into the realm of group actions at our leisure.
Let’s see some examples.
Example5.5.Common Permutation Representations.
For the trivial action, the associated group homomorphism is \(G \to \operatorname{Perm}(S)\) by \(g\mapsto \text{id}_S\)
For left multiplication, the associated group homomorphism is \(G \to \operatorname{Perm}(S)\) by \(g\mapsto \phi_g\text{,}\) where \(\phi_g(s)=gs\)
For the conjugation action, the associated group homomorphism is \(G \to \operatorname{Perm}(S)\) by \(g\mapsto \psi_g\text{,}\) where \(\psi_g(s)=gsg\inv\)
SubsectionFaithful and Transitive Actions
Definition5.6.Faithful.
An action of a group \(G\) on a set \(S\) is called faithful if the associated group homomorphism is injective. Equivalently, an action is faithful if and only if for a given \(g \in G\text{,}\) whenever \(g \cdot s = s\) for all \(s \in S\text{,}\) it must be that \(g = e_G\text{.}\)
Exercise5.7.The Faithful Action.
Verify that the two definitions of a faithful action are indeed equivalent. That is, show that the permutation representation of a group action is injective if and only if for a given \(g \in G\text{,}\) whenever \(g \cdot s = s\) for all \(s \in S\text{,}\) it must be that \(g = e_G\text{.}\)
Discussion5.1.Losing Faith.
Should an action which is not faithful be called unfaithful? Adulterous? Discuss.
Example5.8.Trivial Adultery.
The trivial action is not faithful.
Definition5.9.Transitive.
An action is transitive if for all \(x,y \in S\) there is a \(g \in G\) such that \(x=g\cdot y\text{.}\)
Remark5.10.
In many cases, transitivity and faithfulness are related concepts. As we will see later, if a group action is transitive on a set \(X\text{,}\) then the action is faithful if and only if the stabilizer of any element of \(X\) is trivial. In these situations we can identify the group with a subgroup of the symmetric group on \(X\text{,}\) and use this identification to study the group.
