# Automorphisms of Z^2

1. Feb 25, 2012

### electroweak

I know that the outer-automorphism group of Z^2 is SL(2,Z). Can someone please show me why this is the case? I think Aut(Z^2)=GL(2,Z), but what about Inn(Z^2)? Thanks.

2. Feb 25, 2012

### morphism

How do you know that it's SL(2,Z)? (It's not.)

Note that Z^2 is abelian, so Inn(Z^2) = ??? and consequently Out(Z^2) = ???.

3. Feb 25, 2012

### electroweak

Aut(Z^2)=GL(2,Z), and Inn(Z^2)=Z^2/center(Z^2)=1, so that Out(Z^2)=Aut/Inn=GL(2,Z), right? OK, I figured out what was confusing me; I was applying the Dehn-Neilson theorem (which only holds on hyperbolic surfaces) to the torus (a parabolic surface). This would have equated Out(Z^2) and SL(2,Z). Thanks for confirming my suspicions!