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
.