Finding the right conjugating element of two Sylow-p subgroups.

Question

Sylow's 3rd Theorem did state that the Sylow-p subgroups are conjugates of each other, but is there any general way to find the right element that would show two Sylow-p subgroups are, indeed, conjugates?

Answer 1

Showing the existence of something in math does not mean that we can find it. You may simply think abut pde equations. In most of the cases, we could not find the solution even if we know there is a unique solution.

Sylow thereorems shows that if we have two Sylow $p$-subgroups $P,Q$ of $G$ theren there exists $g\in G$ such that $P^g=Q$. Howovever, it does not tell where this element fits in the group. Yet, by assuming Sylow theorems, all such elements which maps $P$ to $Q$ are exactly the right coset $N_G(P)g$.

There is a generalization of Sylow theorems, known as Alperin fusion theorem. This theroem says where "$g$ might fit in the group". However, both statement and proof of Alperin fusion theorem is not so trivial.

I can leave an easy argument to prove.

If $G$ is a solvable group we can always find such g that whose order is not divisible $p$.