- #1

Kreizhn

- 743

- 1

## Homework Statement

Show that [itex] \text{Aut}_{\text{Grp}}(\mathbb Z /2 \mathbb Z \times \mathbb Z /2 \mathbb Z) \cong S_3 [/itex]. That is, show that the automorphism group of the Klein four group is the symmetric group on 3-letters.

## The Attempt at a Solution

I think the argument here is pretty simple. First we need to analyze the automorphism group. Since automorphisms preserve the identity, there are only 27 possible set mappings. Since automorphisms preserve order and are bijective, there are only 6 possible mappings which correspond to permutations of the 3 non-identity elements, since they all have order 2.

Now one can show by hand that these permutations are all automorphisms. What I am hoping for is a more elegant way of showing this other than doing it by hand. The only thing I can possibly think of is that any two non-identity elements generate the group. Hence so long as we map two distinct elements to any other two distinct elements, we get the entire group. Is this correct? Is there another way of claiming this?