### Math Post the Ninth: Group Actions

Well it looks like that whole "I'll blog during the conference" idea petered out quickly; what with thinking about math enough hours a day to wear me out before blogging, there have been no posts in three weeks. So now, right back into it.

Following some extremely gentle chiding from Neil, along with a link to the following article, I've decided to again defer the promised note on factoring maps and talk instead about group actions, implicitly the subject of the piece linked (you don't need to read it yet).

One of the most fruitful ways to study something is to study its symmetries; correspondingly, one of the most fruitful ways to study groups is to study the things of which they are symmetries. So, how can a group be a symmetry of a thing? Practically every object (ok, every object) out there in the world of mathematics has what's usually called an automorphism (=isomorphism to yourself) group, denoted Aut(X). If X is a set, Aut(X) is composed of the bijections X->X; if X is a group (or other algebraic structure), the isomorphisms; if X is a polygon, the rigid self-maps (rotations and reflections); if X is some kind of bigger geometric object like the plane, symmetries are defined in close to the same way -- as invertible maps which preserve "the structure," where "the structure" is whatever structure you happen to care about at the moment. The simplest way to define a group action of G on a thing X is to say it's a homomorphism \phi:G->Aut(X). Equivalently, if X is a set with some additional structure (like all of our examples thus far), you can write it as a "multiplication" of elements of X by elements of G such (i) ex=x for all x, e the identity of G, and (ii) g(hx)=(gh)x,* as long as that multiplication "respects the structure."

Some examples of group actions include:

(1) A group G acts on the set of its elements by multiplication; just multiply in the group. This isn't a homomorphism, and in fact seems kinda dumb, but is still surprisingly important. Similarly, G acts on the (left) cosets of any subgroup in the same way.

(2) A group G acts on itself as a group by conjugation -- g acting on h is ghg^(-1). This actually is a homomorphism, since if you multiply two g-conjugates together the g's in the middle cancel; thus G acts by conjugation on its set of subgroups. Normal subgroups are precisely those stable under this action. This action is important enough that the subgroup of Aut(G) that is hit by it is called the inner automorphisms and written Inn(G); it is actually a normal subgroup (of Aut(G)!) in its own right, and Aut(G)/Inn(G)=Out(G), the outer automorphisms.

(3) Permutation groups act on the set they're permuting.

(4) Symmetries of polygons act on the polygons.

(5) A subgroup of a group acting on X also acts on X.

With these actions identified, I'll give a few more words. An element x in X has an orbit under G, consisting of all the other elements G can move x to; the subgroup of G which doesn't move x at all is called the stabilizer of x, written Stab_G(x) or myriad other ways that suggest the same thing. An action of G on a set X is called transitive if there is only one orbit; that is, if some element of G can move any element of X to any other. Every action is really a collection of transitive actions (that is, a collection of orbits), and any transitive action is the same thing as the action of G on cosets of the stabilizer of any point in the orbit, which takes some thought but isn't hard to see. An action is faithful if no two elements of G act the same way on X, or the map G->Aut(X) is injective, an action is free if no non-identity element fixes a point (or all stabilizers are trivial), and an action is simply transitive if it is transitive and free; this last implies that G and X have the same number of elements by the point about stabilizers above, since there are no stabilizers in this case. This also implies that a simply transitive action is faithful.

That may seem like a giant load of terminology, but the payoff is Cayley's theorem: every finite group is a subgroup of a permutation group. And in fact with all those words out of the way the proof is almost trivial: G acts on itself by left multiplication, giving a map G->Aut(G) as a set, and Aut(G) as a set is a permutation group. The action is simply transitive, hence faithful, so the map is injective and we're done. The argument here actually generalizes to geometric cases of various kinds, but we won't have to touch those for a long time yet.

Now, go read the article I linked above; it should be easy to figure out where he's going even before he goes there. The exercise to do is this: figure out what the author means by a "golden rule" in terms of group actions. The hint is that it can be said in five words, all of which I've defined already in some math post...

*Writing actions on the left (rather than the mathematically-equivalent right -- that is as gx rather than xg) is actually a slightly bad thing to do if the group G is not Abelian; for reasons that can be avoided if you don't want to dive deeply into the notation, you often want to act by the inverse of some natural action if you're acting on the left. However, it looks slightly more natural for most people, so for informal purposes I'll elide the concern. In fact, this issue is something I've seen distinguished professors argue about at length in class (young, distinguished, argumentative commutative guy didn't care where his action went; old, distinguished, noncommutative but slightly dyslexic guy wanted them on the right), so it should be clear why I'm hoping to skip out on the whole discussion.

Following some extremely gentle chiding from Neil, along with a link to the following article, I've decided to again defer the promised note on factoring maps and talk instead about group actions, implicitly the subject of the piece linked (you don't need to read it yet).

One of the most fruitful ways to study something is to study its symmetries; correspondingly, one of the most fruitful ways to study groups is to study the things of which they are symmetries. So, how can a group be a symmetry of a thing? Practically every object (ok, every object) out there in the world of mathematics has what's usually called an automorphism (=isomorphism to yourself) group, denoted Aut(X). If X is a set, Aut(X) is composed of the bijections X->X; if X is a group (or other algebraic structure), the isomorphisms; if X is a polygon, the rigid self-maps (rotations and reflections); if X is some kind of bigger geometric object like the plane, symmetries are defined in close to the same way -- as invertible maps which preserve "the structure," where "the structure" is whatever structure you happen to care about at the moment. The simplest way to define a group action of G on a thing X is to say it's a homomorphism \phi:G->Aut(X). Equivalently, if X is a set with some additional structure (like all of our examples thus far), you can write it as a "multiplication" of elements of X by elements of G such (i) ex=x for all x, e the identity of G, and (ii) g(hx)=(gh)x,* as long as that multiplication "respects the structure."

Some examples of group actions include:

(1) A group G acts on the set of its elements by multiplication; just multiply in the group. This isn't a homomorphism, and in fact seems kinda dumb, but is still surprisingly important. Similarly, G acts on the (left) cosets of any subgroup in the same way.

(2) A group G acts on itself as a group by conjugation -- g acting on h is ghg^(-1). This actually is a homomorphism, since if you multiply two g-conjugates together the g's in the middle cancel; thus G acts by conjugation on its set of subgroups. Normal subgroups are precisely those stable under this action. This action is important enough that the subgroup of Aut(G) that is hit by it is called the inner automorphisms and written Inn(G); it is actually a normal subgroup (of Aut(G)!) in its own right, and Aut(G)/Inn(G)=Out(G), the outer automorphisms.

(3) Permutation groups act on the set they're permuting.

(4) Symmetries of polygons act on the polygons.

(5) A subgroup of a group acting on X also acts on X.

With these actions identified, I'll give a few more words. An element x in X has an orbit under G, consisting of all the other elements G can move x to; the subgroup of G which doesn't move x at all is called the stabilizer of x, written Stab_G(x) or myriad other ways that suggest the same thing. An action of G on a set X is called transitive if there is only one orbit; that is, if some element of G can move any element of X to any other. Every action is really a collection of transitive actions (that is, a collection of orbits), and any transitive action is the same thing as the action of G on cosets of the stabilizer of any point in the orbit, which takes some thought but isn't hard to see. An action is faithful if no two elements of G act the same way on X, or the map G->Aut(X) is injective, an action is free if no non-identity element fixes a point (or all stabilizers are trivial), and an action is simply transitive if it is transitive and free; this last implies that G and X have the same number of elements by the point about stabilizers above, since there are no stabilizers in this case. This also implies that a simply transitive action is faithful.

That may seem like a giant load of terminology, but the payoff is Cayley's theorem: every finite group is a subgroup of a permutation group. And in fact with all those words out of the way the proof is almost trivial: G acts on itself by left multiplication, giving a map G->Aut(G) as a set, and Aut(G) as a set is a permutation group. The action is simply transitive, hence faithful, so the map is injective and we're done. The argument here actually generalizes to geometric cases of various kinds, but we won't have to touch those for a long time yet.

Now, go read the article I linked above; it should be easy to figure out where he's going even before he goes there. The exercise to do is this: figure out what the author means by a "golden rule" in terms of group actions. The hint is that it can be said in five words, all of which I've defined already in some math post...

*Writing actions on the left (rather than the mathematically-equivalent right -- that is as gx rather than xg) is actually a slightly bad thing to do if the group G is not Abelian; for reasons that can be avoided if you don't want to dive deeply into the notation, you often want to act by the inverse of some natural action if you're acting on the left. However, it looks slightly more natural for most people, so for informal purposes I'll elide the concern. In fact, this issue is something I've seen distinguished professors argue about at length in class (young, distinguished, argumentative commutative guy didn't care where his action went; old, distinguished, noncommutative but slightly dyslexic guy wanted them on the right), so it should be clear why I'm hoping to skip out on the whole discussion.