Say I have a group with presentation like
$$\langle s,t,u \mid s^2,t^2,u^2,(st)^2,(su)^3,(ut)^4\rangle,$$
faithful on set $S$ with exactly one orbit ($|S|$ is known). How could I determine $|\text{Fix}(s)|$, the number of elements fixed by $s$ (equivalently, how many transpositions produce s)? If necessary, suppose I know $|\text{Fix}((st)^n)|$ for all pairs of the generators and for all $n$.
For example, say |S| = 6, and:
- $|\text{Fix}(st)|$ = 0
- $|\text{Fix}(su)|$ = 0
- $|\text{Fix}(tu)|$ = 2
I know that $st$ must be composed of 3 2-cycles (3 transpositions), $su$ is composed of 2 3-cycles (4 transpositions), and $ut$ is composed of 1 4-cycle (3 transpositions). Since $s$, $t$, and $u$ have order 2, they must be products of disjoint transpositions. Therefore, $$|\text{Transpositions}(s)| + |\text{Transpositions}(t)| \geq 3 $$ $$|\text{Transpositions}(s)| + |\text{Transpositions}(u)| \geq 4 $$ $$|\text{Transpositions}(u)| + |\text{Transpositions}(t)| \geq 3 $$ where ${\rm Transpositions}(\sigma)$ is the number of transpositions needed to produce $\sigma$. If you can solve that then $$|\text{Fix}(s)| = n - 2\cdot|\text{Transpositions}(s)|$$ I doubt those thoughts help but that’s as far as I’ve gotten that I’m sure about.