# Isomorphism

Note: ~~ means an isomorphism exists. Moreover,if an isomorphism existed from G to G1 I would say G ~~ G1

Questions: If G is an infinite cyclic group, show that G ~~ Z (Z is the set of integers)

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#### Solution Preview

Proof:

G is an infinite cyclic group, then G=<a>, where a is a generator of G. Now we can define a map f from G to Z as follows:

for any b in G, b=a^i ...

#### Solution Summary

This is a proof regarding isomorphisms and infinite cyclic groups.

Let G be any group, g a fixed element in G. Define phi:G--> G by phi(x) = gxg^-1. Prove that phi is an isomorphism of G onto G.

Group theory

Modern Algebra

Group Theory (LV)

Isomorphism of a Group

Automorphism of a Group

Inner Automorphism of a Group

Let G be any group, g a fixed element in G. Define phi:G--> G by phi(x) = gxg^-1.

Prove that phi is an isomorphism of G onto G.

The fully formatted problem is in the attached file.

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