Sylow-unique prime divisor

This article defines a property of a prime divisor of an integer, one that is important/useful in the use of Sylow theory to study groups

Definition

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.

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:

$n_{p} \equiv 1 mod p$ (the congruence condition in Sylow's theorem)
$n_{p}$ 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