## Conjugation

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.

[...] has a few immediate corollaries. Yesterday, I mentioned the normalizer NG(K) of a subgroup K. When a subgroup H acts on G by conjugation we call the [...]

Pingback by A few more facts about group actions « The Unapologetic Mathematician | February 24, 2007 |

[...] left when you violently rip away the composition from a group and just leave behind its conjugation action. This is a set with an operation , and it’s already a quandle. The part of the [...]

Pingback by Quandles « The Unapologetic Mathematician | March 19, 2007 |

[...] Flag varieties and Lie groups have a really interesting interaction. I’ll try to do the simplest example justice, and the rest are sort of similar. We take a vector space and consider the group of linear transformations with . Clearly this group acts on . If we pick a basis of we can represent each transformation as an matrix. Then there’s a subgroup of “upper triangular” matrices of the form check that the product of two such matrices is again of this form, and that their determinants are always . Of course if we choose a different basis, the transformations in this subgroup are no longer in this upper triangular form. We’ll have a different subgroup of upper triangular matrices. The subgroups corresponding to different bases are related, though — they’re conjugate! [...]

Pingback by Flag varieties « The Unapologetic Mathematician | March 21, 2007 |

[...] most mathematicians have some handle on. We view it as a function of two variables. One runs over the conjugacy classes of the group, while the other runs over a list of (equivalence classes of) irreducible representations. It turns [...]

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[...] basis ? Then we get a new transformation and a new isomorphism of groups . But this gives us an inner automorphism of . Given a transformation , we get the transformation This composite sends to itself, and it [...]

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[...] For some more review, let’s recall the idea of conjugation in a group . We say that two elements and are “conjugate” if for some [...]

Pingback by Conjugates « The Unapologetic Mathematician | September 10, 2010 |

[...] two trivial ideals: and . Another example is the “center” — in analogy with the center of a group — which is the collection of all such that for all . That is, those for which the adjoint [...]

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