Can Z2 x Z3 and D3 Have Isomorphic Automorphism Groups?

[happyg1]

Homework Statement

Give an example of an abelian and a non-abelian group with isomorphic automorphism groups.

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

 

[Hurkyl]

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!

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...

 

[matt grime]

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

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?