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Cayley’s Theorem & Automorphisms (10/16) Cayley’s Theorem. Every group is isomorphic to some permutation group. This says that in some sense permutation groups are “universal” in group theory. If we understood permutation groups completely, then we would understand all abstract groups completely. Sadly, permutation groups tend to be quite complicated!

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A specific implementation of Cayley Let G be an abstract group. Here is one way to get an isomorphism between G and a group of permutations: Given a G, let T a be the permutation of the elements of G created by left multiplication by a (i.e., T a (x) = a x for all x G). We must verify that T a is indeed a permutation. Now G’ = {T a : a G} and let : G G’ be the obvious: (a) = T a. We must verify that is an isomorphism. G’ is called the left regular representation of G. Example: Write down, in cycle notation, the left regular representation of U(10).

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Automorphisms An automorphism of a group G is an isomorphism of G to itself. One automorphism which always exists for any group is the identity automorphism which takes every element to itself. Note that any cyclic group G, any isomorphism from G is completely determined by where it sends a generator. For example, the standard isomorphism from Z to 2Z is set by knowing that 1 goes to 2. So, what are the automorphisms of Z? How about Z 5 ? Z 6 ? How many automorphisms do you think Z n has?

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The group Aut(G) Definition. If G is a group, Aut(G) is the set of all automorphisms of G. Note the Aut(G) always has at least one element. Theorem. For all groups G, Aut(G) is itself a group under function composition. So, what group is Aut(Z) isomorphic to? Want to guess what Aut(Z n ) is isomorphic to? Note: In general, Aut(G) is not easy to determine.

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The group Inn(G) This will seem familiar from the take-home portion of Test #1. Definition. If a G, let a : G G be given by a (x) = a x a -1 for all x G. (“Conjugation by a”) a is called the inner automorphism of G induced by a. Must verify that these are indeed automorphisms of G. Definition. The set of all inner automorphisms of G is denoted Inn(G). Theorem. Inn(G) is itself a group under function composition. Note that if G is abelian, then Inn(G) is just the trivial group. Example: Determine Inn(D 3 ).

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Assignment for Friday Finish reading Chapter 6. Do this carefully as this material is “non-trivial”. On pages 138-9, do Exercises 3, 10, 11, 12, 15, 21, 22, 25.

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