# Groups and Inner Automorphisms

1. Oct 11, 2011

### Locoism

1. The problem statement, all variables and given/known data
Let G be a group. Show that G/Z(G) $\cong$ Inn(G)

3. The attempt at a solution
G/Z(G) = gnZ(G) for some g ε G and for any n ε N
choose some g-1 such that
g(g-1h) = g(hg-1)
and the same can be done switching the g and g-1

This doesn't feel right at all...

2. Oct 11, 2011

### micromass

Staff Emeritus
I don't see what this has to do with the problem??

Can you find a surjective homomorphism

$$f:G\rightarrow Inn(G)$$

and then apply the first isomorphism theorem?

3. Oct 11, 2011

### Dick

No, not right. Given an element g of G can you name an inner automorphism of G corresponding to g? When do two different elements of G, g1 and g2 give define the same automorphism?