Congruence condition on Sylow numbers

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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

Statement

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

n_p \equiv 1 \mod p.

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

[G:N_G(P)] \equiv 1 \mod p.

Related facts

Other parts of Sylow's theorem

n_p | m.

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

Generalizations

Converse

A converse of sorts might be: whenever a is a natural number such that a \equiv 1 \mod p, there exists a finite group G such that n_p = a, i.e., a is the number of p-Sylow subgroups of G.

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

Applications

Proof

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