Recall that for any space $X$, the discrete metric on it is defined by $$d(u,v):=0\ \text{if}\ u=v,$$ $$d(u,v):=1 \ \text{if}\ u\neq v.$$
Now, if $X:=G$ is an abelian group, does the discrete metric on $G$ arise from a norm on $G $?
My first belief is YES, so for $u,v\in G$, I defined $$\|u-v\|:=d(u,v),$$ and I tried to show that $\|\cdot\|$ is a norm.
(1) $\|u\|=d(u,0)$, so $\|u\|=0\iff d(u,0)=0\iff u=0$.
(2) Triangle inequality is also easy to prove. $$\|u+v\|=d(u,-v).$$
Then, if $u=-v$, $$\|u+v\|=0\leq d(u,0)+d(v,0)=\|u\|+\|v\|.$$
On the other hand, if $u\neq -v$, then $u$ and $v$ cannot be $0$ at the same time, so the while the left hand side is $1$, the right hand side is $2$ or $1$.
(3)However, I don't know how to show $\|uv\|=\|u\|\|v\|.$ How could I show this? the multiplication is not even equipped in this group.
Or should I instead show $\|au\|=|a|\|u\|$? but then we need to define the norm over a subfield of a vector space.
Have I got mixed up the definition of norm?
Thank you!
The definition you seem to be attempting to check does not quite make sense, since $G$ is just a group and not a ring, and not a vector space. Arguably, an abelian group is a module over the integers, and it would make sense to define a norm for it as such, but then it can never be bounded in this case, for a nontrivial group, so it would not give rise to the discrete metric.
What would make sense is to define a norm on a group to be a function into the non-negative reals, which is nonzero apart from the identity, and which is submultiplicative (i.e. $\lvert g_1g_2\rvert\leq \lvert g_1\rvert \lvert g_2\rvert$).
Given the way you induce the metric from a norm (which is the only sensible way, I guess), there is not much of a choice for the norm: you must have $\lVert g\rVert=d(e,g)$, which is $0$ at the identity and $1$ everywhere else if $d$ is the discrete metric. It is easy to check that it is submultiplicative. In fact, it has a name. It is called a word norm (where you take the whole $G$ as the generating set).
Note that using this definition, norms on a group correspond exactly to invariant metrics.