Groups and Inner Automorphisms

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SUMMARY

The discussion centers on proving the isomorphism G/Z(G) ≅ Inn(G) for a group G. Participants suggest finding a surjective homomorphism f: G → Inn(G) and applying the first isomorphism theorem as a solution strategy. The conversation highlights the need to identify inner automorphisms corresponding to elements of G and the conditions under which different elements yield the same automorphism. Key concepts include the center of the group Z(G) and inner automorphisms.

PREREQUISITES
  • Understanding of group theory, specifically the concepts of groups and their centers.
  • Familiarity with inner automorphisms and their definitions.
  • Knowledge of the first isomorphism theorem in abstract algebra.
  • Basic skills in constructing homomorphisms between groups.
NEXT STEPS
  • Study the properties of the center of a group, Z(G), and its role in group theory.
  • Learn how to construct inner automorphisms for specific groups.
  • Explore the first isomorphism theorem and its applications in group theory.
  • Investigate examples of groups where G/Z(G) is explicitly calculated and compared to Inn(G).
USEFUL FOR

This discussion is beneficial for students and educators in abstract algebra, particularly those focusing on group theory, as well as mathematicians interested in the structural properties of groups and their automorphisms.

Locoism
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Homework Statement


Let G be a group. Show that G/Z(G) [itex]\cong[/itex] 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...
 
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Locoism said:

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?
 
Locoism said:

Homework Statement


Let G be a group. Show that G/Z(G) [itex]\cong[/itex] 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?
 

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