# 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

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## History

This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page

## 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:

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. Further information: prime divisor greater than Sylow index is Sylow-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. Further information: Order is product of Mersenne prime and one more implies normal Sylow subgroup