The Unapologetic Mathematician

Mathematics for the interested outsider

Conjugation

One of the most useful examples of a group acting on a set comes directly from group theory itself. Let G be a group and H be a subgroup of G. The subgroup H acts on the set of all subgroups of G as follows.

If K is any subgroup of G and h is any element of H, then the set hKh^{-1} of elements of G of the form hkh^{-1} with k in K is another subgroup of G. Indeed, if we take two elements hkh^{-1} and hk'h^{-1} of this set, their product is hkh^{-1}hk'h^{-1}=hkk'h^{-1}, which is again of the proper form since kk' is in K. We call this subgroup the conjugation of K by h.

Given two elements of H we can check that (hh')K(hh')^{-1}=hh'Kh'^{-1}h^{-1}, so conjugating by hh' is the same as conjugating by h', then by h. That is, this defines an action of H on the set of all subgroups of G.

Even better, hKh^{-1} is not just another subgroup of G, it is isomorphic to K. In proving that hKh^{-1} is a subgroup we showed that the function sending k to hkh^{-1} is a homomorphism. We can undo it by conjugating by h^{-1}, so it’s an isomorphism. We say that two subgroups of G related by a conjugation are conjugate.

The subgroup of H sending K to itself — those h in H so that hKh^{-1}=K — is called the normalizer of K in H, written N_H(K). We can verify that K is a normal subgroup in N_G(K), and that K is normal in G exactly when N_G(K)=G.

One orbit is particularly interesting to consider: G is always sent to itself by conjugation. That is, given an element h of G the homomorphism sending g to hgh^{-1} is an automorphism of G. In fact, given any group G, the automorphisms of G themselves form a group, called {\rm Aut}(G). Conjugation gives us a homomorphism c from G to {\rm Aut}(G): given an element g, c(g) is the automorphism of conjugation by g.

We call automorphisms arising in this way “inner automorphisms”. The group {\rm Inn}(G) of inner automorphisms on G is the image of c in {\rm Aut}(G). If g is an element of G and f is any automorphism of G, then f\circ c(g)\circ f^{-1} is the automorphism sending x in G to

f(gf^{-1}(x)g^{-1})=f(g)f(f^{-1}(x))f(g^{-1})=f(g)xf(g)^{-1}

Which is just conjugation of x by f(g). This proves that {\rm Inn}(G) is normal in {\rm Aut}(G). The quotient {\rm Aut}(G)/{\rm Inn}(G) is the group of outer automorphisms {\rm Out}(G).

The kernel of c is the set of elements g so that gg'g^{-1}=g' for all g' in G. That is, for any g' we have gg'=g'g, so the kernel of c is the subgroup of G consisting of elements that commute with every element of G. We call this subgroup the center of G.

Now, consider the group S_n of permutations on n 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.

February 22, 2007 Posted by | Algebra, Group Actions, Group theory | 7 Comments

Rubik’s Magic Cube

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.
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February 22, 2007 Posted by | Group theory, Rubik\'s Cube | 17 Comments