How to tell if two isomorphisms are the same? Let $G$ and $H$ be two groups and there exists two isomorphisms $\phi_1$ and $\phi_2$ between the two groups.
Now, in order to identify two isomorphisms as the same, is it suffice to see what they do to the generating sets? That is, if $S_1$ and $S_2$ are two generating sets of G, and $K_1$ and $K_2$ are of $H$, if $\phi_1(S_1)= \phi_2(S_1) = K_2$ and $S_2$ gets mapped to $K_1$, can I say the two isomorphisms are the same?
Sorry, if this has an obvious answer - I just got exposed to group theory and am trying to learn more.
Thanks!
EDIT: $S_2$ gets mapped to $K_1$ not $K_2$
 A: In you question, it is unclear why you have two generating sets for each group.
But in answer to the big question, yes it is enough to look at what happens to a generating set. But not exactly in the way you are thinking. In particular, if $G$ and $H$ are isomorphic groups, $S$ is a generating set for $G$, and $\phi_{1}$ and $\phi_{2}$ are both isomorphisms from $G \to H$, then the two automorphisms are the same if $\phi_{1}(g) = \phi_{2}(g)$ for all $g \in S$ (you should try to verify that this implies $\phi_{1}(g) = \phi_{2}(g)$ for all $g \in G$).
A: Isomorphism is still a function. Two functions $f_1,f_2\colon A\to B$ equal iff for all $a\in A$ there is $f_1(a) = f_2(a)$.

So, let $G, H$ be groups, $\phi_1,\phi_2\colon G\to H$ isomorphisms homomorphisms. Suppose a set $\{s_i : i\in I\}$ generates $G$, and $\phi_1(s_i) = \phi_2(s_i)$ for all $i\in I$.
Then for an element $a\in G$, by definition of a generating set we have $a = s_{i_1}^{n_1}\cdots s_{i_k}^{n_k}$ for some $k\in\mathbb N$, $i_1,\ldots,i_k\in I$ and $n_1,\ldots,n_k\in\mathbb Z$. Next, $$\phi_1(a) = \phi_1(s_{i_1})^{n_1}\cdots \phi_1(s_{i_k})^{n_k} = \phi_2(s_{i_1})^{n_1}\cdots \phi_2(s_{i_k})^{n_k} = \phi_2(a).$$ Here $a$ was arbitrary, so we conclude $\phi_1 = \phi_2$.
From the proof, it can be seen a set $\{\phi_1(s_i) : i\in I\}$ doesn’t have to generate $H$. Oh it will when $\phi_1$ is iso, but for this proof of equality it’s irrelevant. Also, $\phi_1,\phi_2$ could be simply homomorphisms, not iso.
