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?
Re: Automorphisms of a group G - Inn(G) and Aut(G)
Well, you first need to show that Inn is a subgroup. I am assuming you have done this?
Originally Posted by Bernhard
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 .