One of the most useful examples of a group acting on a set comes directly from group theory itself. Let be a group and be a subgroup of . The subgroup acts on the set of all subgroups of as follows.
If is any subgroup of and is any element of , then the set of elements of of the form with in is another subgroup of . Indeed, if we take two elements and of this set, their product is , which is again of the proper form since is in . We call this subgroup the conjugation of by .
Given two elements of we can check that , so conjugating by is the same as conjugating by , then by . That is, this defines an action of on the set of all subgroups of .
Even better, is not just another subgroup of , it is isomorphic to . In proving that is a subgroup we showed that the function sending to is a homomorphism. We can undo it by conjugating by , so it’s an isomorphism. We say that two subgroups of related by a conjugation are conjugate.
The subgroup of sending to itself — those in so that — is called the normalizer of in , written . We can verify that is a normal subgroup in , and that is normal in exactly when .
One orbit is particularly interesting to consider: is always sent to itself by conjugation. That is, given an element of the homomorphism sending to is an automorphism of . In fact, given any group , the automorphisms of themselves form a group, called . Conjugation gives us a homomorphism from to : given an element , is the automorphism of conjugation by .
We call automorphisms arising in this way “inner automorphisms”. The group of inner automorphisms on is the image of in . If is an element of and is any automorphism of , then is the automorphism sending in to
Which is just conjugation of by . This proves that is normal in . The quotient is the group of outer automorphisms .
The kernel of is the set of elements so that for all in G. That is, for any we have , so the kernel of is the subgroup of consisting of elements that commute with every element of . We call this subgroup the center of .
Now, consider the group of permutations on letters. Determine how this group acts on itself by conjugation. Write out some conjugations in cycle notation to get an idea of what the answer should be.
Rubik’s Cube is a classic fad from the ’80s. Invented by architecture professor Ernő Rubik in 1974 as an illustration of design principles, it became an immensely popular puzzle for a while. Now it’s a hallmark of geekiness — when I visited Dartmouth I saw at least six or seven floating around the graduate student lounge.
What’s most interesting to us here is that it’s also a big case study in group theory. The official website has a pretty good Java implementation, for those who don’t remember (or who have repressed their memories of) the cube. I’ll start talking about the cube after the jump.