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No, "hence det(x) is not an isomorphism between the two groups" (this might be what you meant though). There is a difference between saying that two groups are isomorphic and saying that a particular function is an isomorphism between two groups.But, as you said, many matrices have determinant 1 so the mapping between the two groups is not bijective. Hence it is not isomorphic?

Try constructing the proof yourself (proof by contradiction).You said that because one group is abelian and the other is not indicates that the two groups are not isomorphic, is this another property that I do not know about?

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