Sorry if this is a silly question, but I had a Maths exam today and it asked me to show that 2 groups were isomorphic by "showing one isomorphism" between them.
I simply showed the identity element is isomorphic, but I did not map the other elements in G to H. Was I correct or is one isomorphism a mapping of all the elements?
On
An isomorphism $f:G \to H$ is in particular a function or mapping between the underlying sets. This means that to define $f$ you need to specify an element $f(g) \in H$ for every $g \in G$, not only the identity. Of course you could always define $f(g) = \mathit{identity}_H$ for every $g$, but this so-called trivial homomorphism is never an isomorphism unless both $G$ and $H$ are trivial.
On
The definition of "isomorphic" is an existence statement. Given two groups $G$ and $H$, to say that "$G$ and $H$ are isomorphic" means, by definition, that "There exists an isomorphism $f : G \to H$".
Like any existence statement, the most direct way to prove that $G$ and $H$ are isomorphic is to show that an isomorphism $f$ exists. And to do this requires two steps:
Step 1: Write down a formula for the appropriate function $f : G \to H$ (that's what this problem was asking you to do). In other words, for each $a \in G$ you write down a formula of the form $f(a) = $BLAH BLAH BLAH SOME EXPRESSION INVOLVING THE VARIABLE $a$.
Step 2: Using that formula, prove that $f$ satisfies the definition of "isomorphism": prove it is one-to-one, and it is onto, and $f(ab)=f(a)f(b)$ for all $a,b \in G$.
Very often the hard part of this is Step 1. You cannot, of course, just write down any formula that pops into your head, because if you write down some totally inappropriate formula then most likely it will not be an isomorphism. You must write down an appropriate formula.
So where does the formula come from? It comes from your knowledge and experience. It comes from intuition. It comes from banging your head against the wall. It comes from a long, obsessive hunt. But you cannot avoid it, you must write down a formula. (Well, sometimes you can avoid it, if you find some theorem which guarantees for you that $G$ and $H$ are isomorphic).
What if you just cannot find a formula for an isomorphism, no matter how much knowledge, experience, intuition, head-banging, and hunter's instinct you bring to bear? Well, it's always possible that the groups $G$ and $H$ were not isomorphic to begin with, and so perhaps you should turn the tables and try to prove they are not isomorphic. But that's another story.
I think you should review the definition of isomorphism (and probably homomorphism too). An isomorphism between $G$ and $H$ is a function $f$ from $G$ to $H$ (so, it has to be defined on all elements of $G$, not just the identity of $G$) such that
$f$ is surjective: it hits everything in $H$. (That is, for each $h\in H$ there is some $g\in G$ such that $f(g)=h$.)
$f$ is injective: $f(x)=f(y)$ implies $x=y$.
$f$ is a homomorphism: that is, $f$ respects the group structure. $f(x\cdot_Gy)=f(x)\cdot_Hf(y)$ and $f(x^{-1})=f(x)^{-1}$.
Basically, an isomorphism between $G$ and $H$ is a way of showing that $G$ and $H$ are the "same" group - but maybe with elements named differently. One classic example is the map $$f: (\mathbb{R}, +)\rightarrow (\mathbb{R}_{>0}, \times): x\mapsto e^x,$$ but there are lots of others.
When a problem asks you to "identify an isomorphism between $G$ and $H$," it's asking you to give a specific example of an isomorphism. Isomorphisms are not unique in general; for instance, $$g: (\mathbb{R}, +)\rightarrow (\mathbb{R}_{>0}, \times): x\mapsto 17^x$$ is also an isomorphism.