Let $G=\left<a,b\right>$ be a finite group. What can I say about the order of $G$ with respect to the orders of $a$ and $b$?
Is it true that $|G|\leq |a||b|$, or are there any counterexamples?
What if $\{a,b\}$ forms a minimal generating set for $G$ (i.e. $\left<a\right>$,$\left<b\right>\ne G$)?
If you only know the orders of $a$ and $b$, this gives you basically no upper bound for $|G|$ in general.
For example, a finite group $G$ is dihedral if and only if $G = \langle a,b \rangle$ for $|a| = |b| = 2$. In this case $G$ has order $2n$, where $n = |ab|$.
Also, any finite simple group $G$ is generated by two elements. It is conjectured (and known to be true in many cases) that for a finite simple group $G$ we have $G = \langle a,b \rangle$ for some $a, b \in G$ with $|a| = 2$ and $|b| \in \{3,5,7\}$.
