Groups, Subgroups, and Group Actions, Oh my! (REVISED)

Sorry for the bad title. While I was working on this, it popped into my head, and it won't go away. If I'm going to go crazy hearing that in my head, you can too.

Today's a bit of a definition day. To be able to really show how some of the stuff I've talked about informally works, you need to know some terminology. Don't worry - it's not entirely dull! (I'm also rather overtired this evening, so don't be surprised if some typos slip through; please let me know if you notice any.)

(Update: as the comment above indicated, I originally wrote this in a hurry while overtired. This was a very bad idea. As a result, it has been substantially rewritten to fix it up.)

Subgroups

I've mentioned subgroups briefly once or twice; it's worth pulling it out and making it first class here.

A subgroup S of a group G is a group with the same group operator as G, and whose members are a subset of the members of G. Because the subgroup is also a group, that means that it needs to be closed under the group operation and inverse. To be precise:

A group S is a subgroup of G iff:

members(S) subset members(G)
all x,y in S : x*y in S
all x in S : x^-1 in S

There is a particular kind of subgroup that is interesting, called a normal subgroup. A subgroup S of a group G is a normal subgroup of G if/f:

all x in G : (all y in S : x * y * x^-1 in S)

Another way of writing that is:

all x in G : x * S * x^-1 is a subset of S

It happens to be provable for normal subgroups that the "is a subset of" up there can actually be replaced by "=".

A group is an Abelian Group if/f the group operator is commutative:

all x,y in G : x * y = y * x

If a group is abelian, then all of its subgroups are normal.

A trivial group is a group containing only the identity value.

A simple group is a group whose only normal subgroups are the trivial group and the group itself.

Group actions

We're getting closer to being able to talk about the fun stuff. But we need to talk about group actions.

A group action is something that you can create by treating the members of the group as mappings. The way that you can do this is related to the symmetric groups.

The other day, when I talked about the symmetric groups. I defined the symmetric group of order N as a group over the integers from 1 to N. You can also define the symmetric groups in another way: the symmetric group over a set A is the set of all permutations of the members of the set. The symmetric group over a set A is written S_{A} The two are just different ways of expressing the same basic thing: the group of all permutations over a set of values.

Now, also remember that every group is isomorphic to a subgroup of some symmetry group. What this means is that given an appropriate set A, any group is equivalent to a permutation group over the set. Each member of the group is a function mapping A to A. The group action of a group G is a homomorphism from G to the symmetric group over A, S_{A}.

If you have a group G and a set A, then a group action of G on A is a function f such that:

• all g,h in G : (all a in A : f(g*h,a) = f(g,f(h,a)))
  
• all a in A : f(1,a) = a. (where 1 is the group identity element).
  

That's really a hairy way of saying:

1. There's a mapping from elements of the group to functions over the set;
   
2. Performing the group operation on the elements of set according to the mapping applies a symmetric transformation to the set.
   

Remember when I talked about the symmetries of a pentagon? Rotation, mirroring, etc. Well - for each of those symmetries, there is a group whose operation defines the particular kind of symmetry; and we can define the symmetry on the pentagon figure by applying the group to the set of vertices of the pentagon. For example, there's one group which describes rotational symmetries; for a pentagon, it's a cyclic group (I'll talk about what cyclic groups are later) with 5 elements; each element corresponds to one of the five possible symmetric rotations of the pentagon.