Let $G$ be a finite group. Define $d:G\times G\longrightarrow\mathbb{N}$ by $d(x,y)=o(xy^{-1})-1, \forall\, x,y\in G$. Then $d$ is a metric on $G$ if and only if $$(*)\hspace{5mm}o(ab)<o(a)+o(b), \forall\, a,b\in G.$$It is easy to see that an abelian group satisfies $(*)$ if and only if it is a $p$-group. Moreover, if an arbitrary group $G$ satisfies $(*)$, then all its elements must be of prime power order, i.e. $G$ must be a $CP$-group (a classification of these groups can be found here: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.3738&rep=rep1&type=pdf). I have some difficulties to check whether a $CP$-group satisfies $(*)$. Is there another way to proceed? Also, is it interesting to study such a metric on finite groups?

Additional question: Is it true that all finite groups satisfying $(*)$ are solvable? By the answer below it follows that this class contains no nonabelian simple CP-group (in fact, no nonabelian simple group). Is it possible that a group in the other class (ii) of nonsolvable CP-groups to satisfy $(*)$?