Divisibility 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 states a result of the form that one natural number divides another. Specifically, the (number) of a/an/the (Sylow subgroup) divides the (index) of a/an/the (Sylow subgroup).
View other divisor relations |View congruence conditions

Statement

Suppose $G$ is a finite group of order:

$N=p^{k}m$

where $p$ is prime, $k$ is a nonnegative integer, and $m$ is relatively prime to $p$. Let $n_{p}$ denote the $p$-Sylow number (?), i.e., the number of $p$-Sylow subgroups of $G$. Then we have:

$n_{p}|m$.

Related facts

• Congruence condition on Sylow numbers
• Sylow implies order-conjugate