# Isomorphic automorphisms

1. Jun 11, 2007

### happyg1

1. The problem statement, all variables and given/known data
Give an example of an abelian and a non-abelian group with isomorphic automorphism groups.

3. The attempt at a solution

My classmate talked to our professor and he hints that Z2 xZ3 and D3 (or D6..depends on your notational preference it's the triangle) MIGHT be correct...prove or disprove.....

I see that Z2xZ3 is of order 6 and so is D3. So lovely, off to a good start. At least we start with the same group order.

I need to find the Automorphism group of each set to show that these 2 are isomorphic...(IF they even are) and this is where I can't go any further.

D3 is not abelian but Z2 x Z3 IS abelian and I'm looking at the automorphism groups of each one.

How do I get these automorphism groups? I just am drawing a blank here. We know that each one has the identity Aut, but then how do we define the other ones. We've confused ourselves!

EDIT: So are the automorphisms of D3 $$1, r, r^ 2, a, ra, r^ 2a$$ where 1 is the identity and r is a rotation by 120 degrees and a is a flip through the vertex angle? Or is it something else?

CC

Last edited: Jun 11, 2007
2. Jun 11, 2007

### Hurkyl

Staff Emeritus
The problem is small enough to be solved with sheer brute force.

Each of these groups has 6 elements, and can be presented with two generators. Therefore, there are only thirty-six ways to map the two generators back into the group; you can go through each one and see if it extends to a homomorphism, and if that homomorphism is an automorphism.

After doing the first few, hopefully you'll get some ideas on how to greatly accelerate the search...

3. Jun 12, 2007

### matt grime

Those are the elements of D_2.3, not automorphisms of it. An automorphism of a group is an isomorphism from it to itself.

Auts of Z_2 x Z_3=Z_6 are easy, since that is a cyclic group generated by g say of order 6. Any hom of Z_6 is determined by where it sends g, and there are 6 possiblities - how many of those give isomorphisms?