Let G1 and G2 be groups, and let f: G1 -> G2 be an isomorphism.
If G1 is a cyclic group with generator a, prove that G2 is also a cyclic group, with generator f(a).
Can someone show this proof? Thanks so much!!
Let G1 and G2 be groups, and let f: G1 -> G2 be an isomorphism.
If G1 is a cyclic group with generator a, prove that G2 is also a cyclic group, with generator f(a).
Can someone show this proof? Thanks so much!!
I believe you mean the generator to be $\displaystyle f(a)$. Here's a hint: any element in the first group can be represented as $\displaystyle a^n$ for some integer $\displaystyle n$. Now given an element in the second group (say $\displaystyle b$), we can express $\displaystyle b = f(a^n)$. Can you figure out the rest?