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January 24th, 2016, 01:42 PM   #1
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Group R^× isomorphic to the group R?

Question: Is the group R^× isomorphic to the group R? Why?
R^× = {x ∈ R|x not equal to 0} is a group with usual multiplication as group composition. R is a group with addition as group composition.

Is there any subgroup of R^× isomorphic to R?

What I Know: Sorry, I would have liked to show some steps I took, but Not sure where to begin. I tried, but couldn't get too far. Or well, I can say I know for something to be isomorphic the function should be,
1)Injective
2)Surjective
3)Homomorphism f(ab)=f(a)f(b) for all a,b in group

Do I just show that all elements in real numbers with multiplication defined maps to real numbers has those 3 properties above?

Thank you!!
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January 25th, 2016, 03:55 AM   #2
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To show that two groups are isomorphic you only need to find an isomorphism. But if the two groups are not isomorphic this approach will not work. And as it happens, these groups are not isomorphic.

An isomorphism will carry every subgroup to an isomorphic subgroup. Now you just have to find a subgroup of (R^x,*) that cannot be isomorphic to a subgroup of (R,+).
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