Let $G$ be a finite group admitting a pair of generators $a,b$. I'm happy to assume that $G$ is finite simple.

Sometimes the pair $(a,b)$ is (simultaneously) conjugate to $(a,a^kb)$ for some $k \ge 1$ (ie, there is a $g\in G$ with $gag^{-1} = a, gbg^{-1} = a^kb$). Of course this is always true for $k = |a|$. Let $$k_{a,b} := \min\{k\ge 1\;|\; (a,b) \text{ is (simultaneously) conjugate to }(a,a^kb)\}$$

I'm looking for statements of the form: Under some conditions on $G$, as $(a,b)$ ranges over all generating pairs of $G$, $k_{a,b}$ can't be too small (relative to $|a|$) too often. In particular I'd be interested in an upper bound for the integer $$\text{lcm}\left(\left\{\frac{|a|}{k_{a,b}}\right\}_{a,b}\right)$$ as $(a,b)$ ranges over all generating pairs of $G$.

I'm interested in anything in this direction.

I'm also happy to make an additional restriction: Let $(a,b)$ be a generating pair of a finite simple group $G$. Suppose $a$ has prime order. Could $(a,b)$ be conjugate to $(a,ab)$ by an element of the form $[a^{-1},b^{-1}]^r := (a^{-1}b^{-1}ab)^r$? ($r$ is an integer).

I believe I can show that for $PSL(2,q)$, we always have $k_{a,b} = |a|$, but the proof is quite involved and uses some structure which is special to $PSL(2,q)$. Can we say anything nontrivial for more general finite simple groups?


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