5

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.

Chain Markov
  • 15,564
  • 6
  • 36
  • 116
  • 3
    What is the order of the group constructed by Wall? – the_fox Feb 17 '20 at 14:54
  • 2
    Since you are interested in famous conjectures with large minimal counterexamples, I have added one at this MO-post - in case it is interesting for you. It's about faithful representations of Lie algebras, so sort of closely related to groups via Lazard correspondence. – Dietrich Burde Feb 17 '20 at 16:10
  • 1
    I checked by computer, for $p=5$, there is no example of order $p^7$ or less. – verret Feb 17 '20 at 17:36
  • 1
    @the_fox, the counterexample constructed by Wall has order $5^{46}$ and is simultaneously a counterexample to Hughes conjecture. – Chain Markov Feb 17 '20 at 18:25

0 Answers0