# Congruence condition on Sylow numbers

*This article gives the statement, and possibly proof, of a constraint on numerical invariants that can be associated with a finite group*

This article describes a congruence condition on an enumeration, or a count. It says that in a finite group and modulo prime number, the number of Sylow subgroups, i.e., the number of subgroups of prime power order whose index is relatively prime to the order satisfies a congruence condition.

View other congruence conditions | View divisor relations

## Contents

## Statement

Let be a finite group and a prime. Let be the -Sylow number of , i.e., the number of -Sylow subgroups of . Then:

.

Since all the -Sylow subgroups are conjugate, equals the index of any -Sylow subgroup. Thus, this is equivalent to the following: if is a -Sylow subgroup:

.

## Related facts

### Other parts of Sylow's theorem

- Sylow's theorem: The whole theorem.
- Sylow subgroups exist
- Sylow implies order-conjugate: Any two Sylow subgroups for the same prime are conjugate.
- Sylow implies order-dominating: Any -subgroup is contained in some conjugate of any given -Sylow subgroup.
- Divisibility condition on Sylow numbers: If the order of is where is relatively prime to , then:

.

This fact is often used along with the congruence condition on Sylow numbers.

### Generalizations

- Congruence condition on number of subgroups of given prime power order: Let be a finite group and be any prime power dividing the order of . Then, the number of subgroups of order in is congruent to modulo .
- Congruence condition on Sylow numbers in terms of maximal Sylow intersection: This states that if the intersection of two -Sylow subgroups has index at least , then , the number -Sylow subgroups, is congruent to modulo .
- Congruence condition on number of Sylow subgroups containing a given subgroup of prime power order: This states that if is a finite group and is a -subgroup of , the number of -Sylow subgroups of containing is congruent to modulo .
- Congruence condition on factorization of Hall numbers for a finite solvable group.

### Converse

A converse of sorts might be: whenever is a natural number such that , there exists a finite group such that , i.e., is the number of -Sylow subgroups of .

This is *false*. However, some partial converses are true:

- Converse of congruence condition on Sylow numbers for the prime two: Any odd number occurs as the number of -Sylow subgroups in some finite group.
- Converse of congruence condition for prime power Sylow numbers: If a prime power satisfies the congruence condition, it occurs as a Sylow number.

### Applications

- Prime divisor greater than Sylow index is Sylow-unique
- Order is product of Mersenne prime and one more implies normal Sylow subgroup

## Proof

This proof assumes that we already know that there exist -Sylow subgroups of .