Group Actions

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Example 5.2. Group Actions.

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Trivial Action.
For any group $G$ and any set $S,$ $g \cdot s := s$ defines an action, the trivial action.
The group $S_{n}$ acts on the set $X = {1, 2, \dots, n}$ such that $σ \cdot i = σ (i) .$
Left Multiplication.
For any group $G$ and any set $S,$ $g \cdot s := g s$ 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 $S,$ $g \cdot s := g x g^{- 1}$ defines an action, the conjugation action.

Answer.

Let

G

be a group and

S

a set.

Let $g, g^{'} \in G$ and $s \in S .$ Notice $g \cdot (g^{'} \cdot s) = g \cdot s = s$ and $e_{G} \cdot s = s .$ Thus the trivial action is indeed a group action.

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This is a small assemblage with no title, simply to make sure the surrounding box behaves properly, especially for LaTeX output.

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Remark 5.3.

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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.

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Subsection The Permutation Representation

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One can also think of group actions as homomorphisms.

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Theorem 5.4. Permutation Representation.

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Assume

(G, \cdot)

is a group and

S

is a set.

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If $\cdot$ is an action of $G$ on $S,$ then the function $ρ : G \to Perm (S)$ defined as $ρ (g) = σ_{g},$ where $σ_{g} : S \to S$ is the function given by $σ_{g} (s) = g \cdot s,$ is a well defined homomorphism of groups.
Conversely, if $ρ : G \to Perm (S)$ is a group homomorphism, the rule $g \cdot s := ρ (g) (s)$ defines an action of $G$ on $S .$

Proof.

We have

\begin{aligned} (σ_{g} \circ σ_{g^{- 1}}) (s) & = σ_{g} (σ_{g^{- 1}} (s)) & (def of composition) \\ = g \cdot (g^{- 1} \cdot s) & (def for σ_{g} and σ_{g^{- 1}}) \\ = (g g^{- 1}) \cdot s & (first property of a group action) \\ = e_{G} \cdot s & (group axiom) \\ = s & (second property of a group action) \end{aligned}

thus

σ_{g} \circ σ_{g^{- 1}} = {id}_{S}

and a similar argument shows that

σ_{g^{- 1}} \circ σ_{g} = {id}_{S}

Finally, we wish to show

ρ (g g^{'}) = ρ (g) \circ ρ (g^{'}),

equivalently

σ_{g g^{'}} = σ_{g} \circ σ_{g^{'}} .

Since

σ_{g g^{'}} (s) = (g g^{'}) \cdot s = g \cdot (g^{'} \dots) = σ_{g} (σ_{g^{'}} (s)) = (σ_{g} \circ σ_{g^{'}}) (s),

holds for all

s,

this proves

ρ

is a homomorphism.

Given a homomorphism

ρ,

the function

G \times S \to S

defined as

g \cdot s = ρ (g) (s)

is an action because

g^{'} (g s) = ρ (g^{'}) (ρ (g) (s)) = (ρ (g^{'}) \circ ρ (g)) (s) = ρ (g g^{'}) (s) = (g g^{'}) s,

and

e_{G} s = ρ (e_{G}) (s) = id (s) = s .

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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.

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Let’s see some examples.

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Example 5.5. Common Permutation Representations.

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For the trivial action, the associated group homomorphism is $G \to Perm (S)$ by $g \mapsto {id}_{S}$
For left multiplication, the associated group homomorphism is $G \to Perm (S)$ by $g \mapsto ϕ_{g},$ where $ϕ_{g} (s) = g s$
For the conjugation action, the associated group homomorphism is $G \to Perm (S)$ by $g \mapsto ψ_{g},$ where $ψ_{g} (s) = g s g^{- 1}$

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Subsection Faithful and Transitive Actions

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Definition 5.6. Faithful.

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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,

whenever

g \cdot s = s

for all

s \in S,

it must be that

g = e_{G} .

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Exercise 5.7. The Faithful Action.

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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,

whenever

g \cdot s = s

for all

s \in S,

it must be that

g = e_{G} .

Discussion 5.1. Losing Faith.

Should an action which is not faithful be called unfaithful? Adulterous? Discuss.

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Example 5.8. Trivial Adultery.

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The trivial action is not faithful.

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Definition 5.9. Transitive.

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An action is transitive if for all

x, y \in S

there is a

g \in G

such that

x = g \cdot y .

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Remark 5.10.

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In many cases, transitivity and faithfulness are related concepts. As we will see later, if a group action is transitive on a set

X,

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,

and use this identification to study the group.

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Summary

One of the central topics of group theory is the Group Action. The two main group actions we will examine are that of Left Multiplication and Conjugation.
Through the Permutation Representation group actions can be viewed as homomorphisms. This will be used extensively.
Group actions can be both Faithful and Transitive.

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