Abstract Algebra/Group Theory/Group actions on sets

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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[edit]

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 such 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[edit]

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.