Sylow-unique prime divisor

Symbol-free divisor
A prime divisor of a number is said to be Sylow-unique if for every group whose order is that number, there is a unique Sylow subgroup corresponding to that prime divisor.

Definition with symbols
A prime divisor $$p$$ of a number $$N$$ is said to be Sylow-unique if for any group $$G$$ of order $$N$$, there is a unique $$p$$-Sylow subgroup.

Stronger properties

 * Sylow-direct prime divisor: A prime divisor such that the Sylow subgroup for that prime divisor is a direct factor of that group, for any group of the given order

Weaker properties

 * Core-nontrivial prime divisor
 * Closure-proper prime divisor

Testing for Sylow-uniqueness
We fix some notation. Let $$p$$ be the prime divisor, $$k$$ the exponent of $$p$$ in $$N$$, and $$m$$ the coprime part, viz $$m = N/p^k$$.

Divisibility and congruence tests
Let $$n_p$$ denote the number of $$p$$-Sylow subgroups in the given group $$G$$. We know that the following hold:


 * congruence condition on Sylow numbers: $$n_p \equiv 1 \mod p$$ (the congruence condition in Sylow's theorem)
 * divisibility condition on Sylow numbers: $$n_p$$ divides $$m$$ (the divisibility condition)

Note that both these conditions are purely in terms of $$N$$ and $$p$$ and do not depend on $$G$$. if the only solution to both these conditions is the solution $$n_p = 1$$, then clearly, $$p$$ is Sylow-unique.

Note, however, that while a unique solution to the congruence and divisibility conditions guarantees Sylow-uniqueness, the converse is not true. This is because there may be solutions to the congruence and divisibility conditions that do not get realized for actual groups.

Some special cases
One special case where Sylow-uniqueness can be guaranteed is when the prime divisor is greater than the index $$m$$. In other words, if $$N = p^km$$ where $$m < p$$, then the $$p$$-Sylow subgroup in any group of order $$N$$ is unique.

Sometimes, close variants of this method guarantee that there exists some nontrivial normal Sylow subgroup, although we cannot ascertain which prime divisor it corresponds with.

Presentations/talks on this

 * Using Sylow theory in the classification of finite simple groups