There existed once a “folklore” conjecture that stated:
Suppose $p$ is a prime. Then any finite group $G$ with $> (1 - \frac{p-1}{p^2})|G|$ elements of order $p$ has exponent $p$
This conjecture was disproved by G.E.Wall in 1965, who constructed a group of exponent $25$, where all elements of order $25$ lie in the subgroup of index $25$.
My question is:
What is the minimal possible size of a counterexample to this conjecture?
I am interested in it because I collect disproven conjectures with large minimal counterexamples
Note, that $p$ in this counterexample has to be $5$ of greater, as the conjecture was proven to be true for $p = 2$ and $p = 3$ by Thomas Laffey in 1976.