Automorphisms of a group G - Inn(G) and Aut(G)

Dummit and Foote Section 4.4 Automorphisms Exercise 1 reads as follows:

Let and is conjugation by g, prove that = .

**Deduce that Inn(G) is a normal subgroup of Aut(G)**

================================================== ====

A start to the proof is as follows:

=

=

=

=

Thus = .

My question is - **how do we use this result to show Inn(G) is a normal subgroup of Aut(G)?**

Can someone please help?

Peter

Re: Automorphisms of a group G - Inn(G) and Aut(G)

Quote:

Originally Posted by

**Bernhard** Dummit and Foote Section 4.4 Automorphisms Exercise 1 reads as follows:

Let

and

is conjugation by g, prove that

=

.

**Deduce that Inn(G) is a normal subgroup of Aut(G)**
================================================== ====

A start to the proof is as follows:

=

=

=

=

Thus

=

.

My question is -

**how do we use this result to show Inn(G) is a normal subgroup of Aut(G)?**
Can someone please help?

Peter

Well, you first need to show that Inn is a subgroup. I am assuming you have done this?

Now, a normal subgroup, , is a subgroup whhere for all and . So...you took an arbitrary automorphism, , and an arbitrary inner automorphism, , and showed that which is precisely what you want. Here, , , and .