Diameter of Cayley graph of a finite group on the p-singular elements for some prime p

Let be a finite group and be a prime dividing . Let be the set of -singular elements of i.e., . Here is a conjecture of min.

Conjecture. If is a solavble group then .

Here, as usual, for a non-empty susbet of a group.

It is not hard to prove if is -nilpotent group then the conjecture is valid.

I have checked for solvable groups of order 96, 1000 and 2000 that the conjecture is true.

To study a minimal counterexample, one may assume that .

I have also proposed the conjecture in MatheOverflow

https://mathoverflow.net/questions/504343/diameter-of-p-singular-cayley-graph-of-a-finite-group

I am ready to collaborate on my conjecture if someone interested in. One may reach me via [email protected]