The answer has no details. Hence maybe the answer is supposed to be quick. But I can't see it?
Hence I took two groups. Call them $G_1 = \{a, b, c\}, G_2 = \{d, e, f\}$.
Then because every group has an identity, I know $G_1, G_2$ has one each.
Hence WLOG pick $c$ as the identity in $G_1$. I want to match letters so pick $e$ as the identity in $G_2$.
Now we have $G_1 = \{a, b, \color{magenta}{c}\}, G_2 = \{d, \color{magenta}{e}, f\}$.
I know every group has an inverse. But how do I apply this to $G_1, G_2$ to simplify them?
And to prove $G_1, G_2$ are isomorphic, how do I envisage and envision what the isomorphism is?
Update Dec. 25, 2013 (1). Answer from B.S. Why does $ab = b$ fail?
$\begin{align} ab & = b \\ & = bc \end{align}$. What now?
(2.) Do I have to do all the algebra work for $G_1$ for $G_2$? Is there some smart answer?
Update Jan. 8, 2014 (1.) I'm confounded by drhab's comment on Dec. 30 2013. Is drhab saying: Even if the domain's identity is the $\color{green}{third}$ letter $\color{magenta}{c}$ but the codomain's identity is the second letter $\color{magenta}{e}$, $d^{\huge{\color{green}{3}}} = e$ anyways? Hence I should've chosen the $\color{green}{third}$ letter in the codomain as the identity too?
What else is drhab saying about this?
(2.) I don't understand drhab's comment on Dec. 28 2013. Why refer to commutativity? It's not a group axiom? And what are the binary operations?
Update: I didn't realize this before, but by dint of Martin Sleziak's comment, this question is just a sepcial case of Fraleigh p. 63 Theorem 6.10 = Pinter p. 109-111 Theorem 11.1.
Best Answer
Let's work on $G_1$ and assume $c=e_{G_1}$. So $$ac=a, bc=b,cc=c$$ But what about $ab$. It is either $a$, $b$ or $c$. If $ab=a=ac$, since $G_1$ is a group so we can cancel $a$ to have $b=c$ which is wrong because $b\neq c$. The same short story is when we assume $ab=b$, so we just have $ab=c$. This means that $$a=b^{-1}$$ and vise versa. Now what about $aa$? Is it equal to $a$, $b$ or $c$? We have $$aa=a\to a=c\\ aa=b\to aa=a^{-1}\to a^3=c\\ aa=c\to a=a^{-1}=b$$ The first one and the last one is clearly wrong so we just get $a^3=c$. Hence our group $G$ is change to the following form: $$G_1=\{c,a,a^{-1}\}$$ in which $a^3=c$. Now do the same way for $G_2$. It gives us: $$G_2=\{e,d,d^{-1}\}$$ What are the differences between these two groups? Just changing the alphabet $a\to d$ and vice versa? So, there are no differences between them and they are really the same things.