Groups and Inner Automorphisms

Locoism

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

micromass

Quote by Locoism

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...

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

Can you find a surjective homomorphism

[tex]f:G\rightarrow Inn(G)[/tex]

and then apply the first isomorphism theorem?

Dick

Quote by Locoism

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

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?

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Locoism	Groups and Inner Automorphisms Tweet 1. The problem statement, all variables and given/known data Let G be a group. Show that G/Z(G) [itex]\cong[/itex] 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...