# Groups and Inner Automorphisms

by Locoism
Tags: automorphism, centraliser, group, subgroup
 P: 81 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...
Mentor
P: 18,334
 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

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

and then apply the first isomorphism theorem?
 Quote by 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...