How would I show that two groups are isomorphic? 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.