Groups and Inner Automorphisms

[Locoism]

Homework Statement

Let G be a group. Show that G/Z(G) $\cong$ Inn(G)

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...

 

[micromass]

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...

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?

 

[Dick]

Homework Statement

Let G be a group. Show that G/Z(G) $\cong$ Inn(G)

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...

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?