Groups and Inner Automorphisms


by Locoism
Tags: automorphism, centraliser, group, subgroup
Locoism
Locoism is offline
#1
Oct11-11, 09:38 PM
P: 81
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...
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micromass
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#2
Oct11-11, 10:10 PM
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Quote Quote by Locoism View Post
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
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#3
Oct11-11, 10:18 PM
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Thanks
P: 25,163
Quote Quote by Locoism View Post
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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