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FOR EXAMPLE:

Take the group homomorphism φ : ((0, oo), x) → ((0, oo), x) defined by φ (x) = x²

Since φ is taking any element in (0, oo) and operating on it by x, does it map one-to-one and onto to (0, oo)?

I assume by showing that two groups are isomorphic you have to show that there is a one-to-one correspondence and that they are onto (ie. the two groups are a bijection).

Would I start by taking some element a of ((0, oo), x) and then say that under x, a is mapped to a². Then for all a, a² is in (0, oo) hence it is one-to-one. Then show that there is only one a that maps to a² in (0, oo) hence it is onto. Since it is both then it is isomorphic.

Any help would be appreciated.