Disproving Isomorphism of G/N & G'/N': Counterexample Needed

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SUMMARY

The discussion centers on disproving the isomorphism of the quotient groups G/N and G'/N' under the conditions that N is a normal subgroup of G, N' is a normal subgroup of G', and G is isomorphic to G'. The key counterexample involves the subgroups of integers, specifically the subgroup of even integers, which is isomorphic to the integers themselves. The hint provided indicates that the subgroup { ... -4, -2, 0, 2, 4 ... } (even integers) is isomorphic to the subgroup { ... -2, -1, 0, 1, 2, ... } (all integers), demonstrating that G/N is not necessarily isomorphic to G'/N'.

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  • Understanding of group theory concepts, specifically normal subgroups.
  • Familiarity with isomorphism in the context of algebraic structures.
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  • Basic understanding of integer subgroups and their relationships.
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This discussion is beneficial for students of abstract algebra, particularly those studying group theory, as well as mathematicians seeking to deepen their understanding of isomorphism and quotient groups.

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Prove or disprove:

Suppose N is a normal subgp of G and N' is a normal subgp of G'. If G is isomorphic to G' and N is isomorphic to N' does that mean that G/N is isomorphic to G'/N'?

I was trying to work out a proof until my professor told us to think of subgroups of the integers when doing this problem. So now I'm trying to disprove it through a counterexample. I have been stuck on this question for a while and I would appreciate any help.

Thank you.
 
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Hint: { ... -4, -2, 0, 2, 4 ... } is isomorphic to { ... -2, -1, 0, 1, 2, ... }.
 

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