# Fifth Sylow Theorem

## Theorem

The number of Sylow $p$-subgroups of a finite group is a divisor of their common index.

## Proof 1

By the Orbit-Stabilizer Theorem, the number of conjugates of $P$ is equal to the index of the normalizer $\map {N_G} P$.

Thus by Lagrange's Theorem, the number of Sylow $p$-subgroups divides $\order G$.

Let $m$ be the number of Sylow $p$-subgroups, and let $\order G = k p^n$.

From the Fourth Sylow Theorem, $m \equiv 1 \pmod p$.

So it follows that $m \nmid p \implies m \nmid p^n$.

Thus $m \divides k$ which is the index of the Sylow $p$-subgroups in $G$.

$\blacksquare$

## Proof 2

Let $G$ be a finite group of order $p^n m$, where $p \nmid m$ and $n > 0$.

Let $r$ be the number of Sylow $p$-subgroups of $G$.

It is to be shown that $r \divides m$.

Let $H$ be a Sylow $p$-subgroup of $G$.

We have that:

- $\order H = p^n$

- $\index G H = m$

Let $S_1, S_2, \ldots, S_m$ denote the elements of the left coset space of $G / H$.

We have that $G$ acts on $G / H$ by the rule:

- $g * S_i = g S_i$

for $S_i \in G / H$.

There is only one orbit under this action, namely the whole of $G / H$.

Therefore the stabilizer of each $S_i$ is a subgroup of $G$ of index $m$ and order $p^n$.

In other words, each $S_i$ has a Sylow $p$-subgroup as a stabilizer.

Now it is shown that each Sylow $p$-subgroup is the stabilizer of one or more of the cosets $S_1, S_2, \ldots, S_m$.

We have that $H$ is the stabilizer of the coset $H$, which must be one of $S_1, S_2, \ldots, S_m$.

Let $S_1, S_2, \ldots, S_k$ be the elements of $G / H$ whose stabilizer is $H$.

By the Third Sylow Theorem, any other Sylow $p$-subgroup of $H$ is a conjugates $g H g^{-1}$ of $H$.

Thus it is seen that $g H g^{-1}$ is a stabilizer of the cosets $g S_1, g S_2, \ldots, g S_k$.

So each of the $r$ distinct Sylow $p$-subgroups of $G$ is the stabilizer of exactly $k$ elements of $G / H$.

Thus:

- $m = k r$

and so:

- $r \divides m$

as required.

$\blacksquare$

## Also known as

Some sources call this the **fourth Sylow theorem** and merge it with what we call the Fourth Sylow Theorem.

Others merge this result with what we call the Fourth Sylow Theorem and call it the **third Sylow theorem**.

Others merge this with what we call the Third Sylow Theorem and call it the **third Sylow theorem**.

## Also see

## Source of Name

This entry was named for Peter Ludwig Mejdell Sylow.

## Historical Note

When cracking open the structure of a group, it is a useful plan to start with investigating the prime subgroups.

The **Sylow Theorems** are a set of results which provide us with just the sort of information we need.

Ludwig Sylow was a Norwegian mathematician who established some important facts on this subject.

He published what are now referred to as the **Sylow Theorems** in $1872$.

The name is pronounced something like **Soolof**.

There is no standard numbering for the **Sylow Theorems**.

Different authors use different labellings.

Therefore, the nomenclature as defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ is to a greater or lesser extent arbitrary.