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

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In summary, the question is whether if N is a normal subgroup of G and N' is a normal subgroup of G', and if G is isomorphic to G' and N is isomorphic to N', does that mean that G/N is isomorphic to G'/N'? The conversation includes the speaker's attempt to prove the statement and their professor's suggestion to think about subgroups of integers. The speaker is now trying to disprove the statement with a counterexample and is seeking help. The hint given is that the set { ... -4, -2, 0, 2, 4 ... } is isomorphic to { ... -2, -1, 0, 1, 2, ... }.
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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, ... }.
 

What is isomorphism?

Isomorphism is a mathematical concept that describes a relationship between two objects that preserves their structure and properties. In other words, two objects are isomorphic if they have the same underlying structure, even if their individual elements may appear different.

What is G/N and G'/N'?

G/N and G'/N' are known as quotient groups and are used in group theory to describe the relationship between a group G and one of its normal subgroups N. They represent the cosets, or subsets, of G that are formed by dividing the elements of G by the elements of N. G'/N' is a different quotient group that is formed using a different normal subgroup of G.

How can I disprove isomorphism of G/N and G'/N'?

To disprove isomorphism, you would need to find a counterexample that shows that the two quotient groups are not isomorphic. This means finding a difference in structure or properties between the two groups that cannot be preserved through an isomorphism.

What makes a good counterexample?

A good counterexample should clearly demonstrate a difference between the two quotient groups G/N and G'/N' that cannot be preserved through an isomorphism. This can be done by showing a difference in structure, such as different orders or number of elements, or a difference in properties, such as different commutativity or associativity.

Why is disproving isomorphism important?

Disproving isomorphism is important because it helps in understanding the underlying structure and properties of mathematical objects. It also allows us to identify and correct errors in mathematical reasoning and can lead to new insights and discoveries in the field of mathematics.

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