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.

Relation with other properties

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:

• $n_{p}\equiv 1\mod p$ (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 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.