# Abstract Algebra/Group Theory/Group actions on sets

In this section, we will encounter most important application of group theory. This is the notion of an action on some object. Over-generalizing slightly, we only care about groups because they act on things.

## Group Actions

There are two equivalent ways to define a group action. We will present both, then prove their equivalence.

Definition 1a: Let $G$ be a group and $X$ a set. Then a group action (or just action) of $G$ on $X$ is a binary operation $*\,:\,G\times X\rightarrow X$ such that for any $g,h\in G$ and any $x\in X$,

i) $e*x=x$
ii) $g*(h*x)=(gh)*x$

Definition 1b: Let $G$ be a group and $X$ a set. Then a group action (or just action) of $G$ on $X$ is a homomorphism $\sigma\,:\, G\rightarrow S_X$.

Theorem 2: Definitions 1a and 1b are equivalent.

Proof: We must show that for each binary operations in Definition 1a there is a unique homomorphism as in Definition 1b and vice versa. Given the binary operation, define the permutation $\sigma(g)\in S_X$ given by $\sigma(g)(x)=g*x$. Then $\sigma(g)\circ\sigma(h)(x)=\sigma(g)(\sigma(h)(x))=g*(h*x)=(gh)*x=\sigma(gh)(x)$ for all $x\in X$ and $g,h\in G$. Thus $\sigma$ is a homomorphism $G\rightarrow S_X$. Now, given a homomorphism $\sigma$, define a binary operation $*\,:\, G\times X\rightarrow X$ by $g*x=\sigma(g)(x)$. Then $e*x=\sigma(e)(x)=\mathrm{id}_X(x)=x$ and $g*(h*x)=\sigma(g)\circ\sigma(h)(x)=\sigma(gh)(x)=(gh)*x$. Thus $*$ is a binary operation satisfying the axioms in Definition 1a, and we are done.

Example 3: Let $G$ be a group. Then $G$ acts on itself by left multiplication, that is, $g*h=gh$ for any $g,h\in G$.

Example 4: Let $G$ be a group with $H$ a any subgroup. Then $G$ acts on the left cosets $G/H$ by left multiplication, that is, $g*(g^\prime H)=(gg^\prime)H$ for any $g,h\in G$.

Definition 5: A set which is acted on by a group $G$ is called a $G$-set.

Definition 6: Let $G$ be a group acting on a set $X$. Then the action of $G$ is said to be

i) faithfull if $e$ is the only element in $G$ such that $g*x=x$ for all $x\in X$, and
ii) free if $e$ is the only element in $G$ such that $g*x=x$ for any element in $X$.

Remark 7: A group acting on itself by left multiplication is a free action. Thus is it also faithfull.

Definition 8: Let $X$ be a $G$-set. The action is called transitive if for any $x,y\in X$, there exists a $g\in G$ sich that $y=g*x$.

Remark 9: We see that the actions in Example 3 and Example 4 are transitive.

Definition 10: Let $X$ be a $G$-set and $x\in X$. Define the orbit of $x$ to be the set $G*x=\{g*x\mid g\in G\}$.

Definition 11: Let $X$ be a $G$-set and $x,y\in X$. Define the equivalence relation $x\sim y \,\Leftrightarrow\, (\exist g\in G) y=g*x \,\Leftrightarrow\, y\in G*x$. (Check that this is an equivalence relation!) Then $X/\sim$ is called the orbit space of $X$ with respect to $G$, and we write $X/\sim \equiv X/G$.

Remark 12: Since the action is obviously transitive on each orbit, to "understand" group actions we only have to understand transitive actions.

Definition 13: Let $X$ be a $G$-set. The stabilizer of $x\in X$ is the set $G_x=\{g\in G\mid g*x=x\}$.

Lemma 14: Let $X$ be a $G$-set and let $x\in X$. Then $G_x$ is a subgroup of $G$.

Proof: Since $e*x=x$, $e\in G_x$. Let $a,b\in G_x$. Then $(ab)*x=a*(b*x)=a*x=x$, so $ab\in G_x$. Finally, $x=e*x=(b^{-1}b)*x=b^{-1}*(b*x)=b^{-1}*x$, so $b^{-1}\in G_x$. Thus $G_x$ is a subgroup of $G$.

Lemma 15: Let $X$ be a $G$-set, $a\in X$, $g\in G$ and $b=g*a\in X$. Then $G_b=gG_ag^{-1}$.

Proof: Let $h\in G_a$. Then $(ghg^{-1})*b=(ghg^{-1}g)*a=(gh)*a=g*(h*a)=g*a=b$, showing $gG_ag^{-1}\subseteq G_b$. To show $G_b\subseteq gG_ag^{-1}$, note that $a=g^{-1}*b$ and follow the same arugment, obtaining $g^{-1}G_bg\subseteq G_a$.

A $G$-set is an algebraic structure, and as in any situation where we have several instances of a structure, we cannot resist the temptation to introduce maps between them.

Definition 16: Let $X$ and $X^\prime$ be $G$-sets with binary operations $*$ and $*^\prime$ respectively. Then a function $f\,:\, X\rightarrow X^\prime$ is called an equivariant function if $f(g*x)=g*^\prime f(x)$ for all $x\in X$. $f$ is called an isomorphism if $f$ is bijective.

Lemma 17: Compositions of equivariant functions are equivariant.

Proof: Trivial.

Theorem 18: Every transitive action of $G$ on a set $X$ is isomorphic to left multiplication on $G/H$, where $H=G_x$ for any $x\in X$.

Proof: By Lemma 15, the stabilizers of any two elements in an orbit are conjugate and so isomorphic. Thus the chosen element is arbitrary. Let $f\,:\, G/H\rightarrow X$ by $f(gH)=g*x$ for all $g\in G$. Then if $g_1H=g_2H$, we have $g_1g_2^{-1}\in H$, so $(g_1g_2^{-1})*x=x$ and $g_1*x=g_2*x$ and so $f$ is well-defined. Now let $h\,:\, X\rightarrow G/H$ be defined by $h(g*x)=gH$. If $g_1*x=g_2*x$, then $(g_1g_2^{-1})*x=x$, so $g_1g_2^{-1}\in H$ and $g_1H=g_2H$ showing that $h$ is well-defined. Since $f$ and $h$ are obviously inverses of each other, this shows that $f$ is a bijection. To prove equivariance, observe that $f(g_1gH)=(g_1g)*x=g_1*(g*x)=g_1*f(gH)$.

Corollary 19 (orbit stabilizer theorem): Let $X$ be a $G$-set and $G*x$ the orbit of $x\in X$. Then $|G*x||G_x|=|G|$.

Proof: Indeed, from Theorem 18 and Lagrange's theorem we obtain $|G*x||G_x|=|G|$.

Definition 20: The orbit of $x\in X$ where $X$ is a $G$-set is said to be trivial if $G*x=\{x\}$. Let $Z$ denote the set of elements of $X$ whose orbits are trivial. Equivalently, $Z=\{x\in X\mid (\forall g\in G) g*x=x\}$.

Lemma 21: Let $S$ be a finite set and $G$ be a group acting on $S$. Next, let $A$ be a set containing exactly one element from each nontrivial orbit of the action. Then, $|S|=|Z|+\sum_{a\in A} |G/G_a|$.

Proof: Since the orbits partition $S$, the cardinality of $S$ is the sum of cardinalities of the orbits. All the trivial orbits are contained in $Z$, so $|S|=|Z|+\sum_{a\in A} |G*a|$. By Corollary 19, $|G*a|=|G/G_a|$, proving the lemma.

Definition 22: A $p$-group is a group whose order is a finite power of a prime integer $p$.

Corollary 23: Let $G$ be a $p$-group acting on a set $S$. Then $|S|\equiv |Z|\ \mathrm{mod}\ p$.

Proof: Since $G$ is a $p$-group, $|G*a|$ divides $p$ for each $a\in A$ with $A$ defined as in Lemma 21. Thus $\sum_{a\in A}|G*a|\equiv 0\ \mathrm{mod}\ p$.

## Group Representations

Linear group actions on vector spaces are especially interesting. These have a special name and comprise a subfield of group theory on their own, called group representation theory.　We will only touch slightly upon it here.

Definition 24: Let $G$ be a group and $V$ be a vector space over a field $F$. Then a representation of $G$ on $V$ is a map $\Phi\,:\, G\times V\rightarrow V$ such that

i) $\Phi(g)\,:\, V\rightarrow V$ given by $\Psi(g)(v)=\Psi(g,v)$, $v\in V$, is linear in $v$ over $F$.
ii) $\Phi(e,v)=v$
iii) $\Phi\left(g_1,\Phi(g_2,v)\right)=\Phi(g_1g_2,v)$ for all $g_1,g_2\in G$, $v\in V$.

V is called the representation space and the dimension of $V$, if it is finite, is called the dimension or degree of the representation.

Remark 25: Equivalently, a representation of $G$ on $V$ is a homomorphism $\phi\,:\, G\rightarrow GL(V,F)$. A representation can be given by listing $V$ and $\phi$, $(V,\phi)$.

As a representation is a special kind of group action, all the concepts we have introduced for actions apply for representations.

Definition 26: A representation of a group $G$ on a vector space $V$ is called faithful or effective if $\phi\,:\, G\rightarrow GL(V,F)$ is injective.