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Cayley’s Theorem & Automorphisms (10/16)

Cayley’s Theorem & Automorphisms (10/16). Cayley’s Theorem . Every group is isomorphic to some permutation group.

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Cayley’s Theorem & Automorphisms (10/16)

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  1. 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!

  2. 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 Tabe the permutation of the elements of G created by left multiplication by a (i.e., Ta(x) = a x for all x  G). We must verify that Ta is indeed a permutation. • Now G’ = {Ta: a G} and let : G  G’ be the obvious:(a) = Ta. 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).

  3. 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 to2Z is set by knowing that 1 goes to 2. • So, what are the automorphismsof Z? • How about Z5? Z6? • How many automorphisms do you think Zn has?

  4. 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(Zn) is isomorphic to? • Note: In general, Aut(G) is not easy to determine.

  5. 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) = ax 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(D3).

  6. 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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