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!

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 Ta be 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).

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 Z5? Z6?

• How many automorphisms do you think Zn has?

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.

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(D3).

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.