Third Sylow Theorem

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Theorem

All the Sylow $p$-subgroups of a finite group are conjugate.


Proof 1

Suppose $P$ and $Q$ are Sylow $p$-subgroups of $G$.

By the Second Sylow Theorem, $Q$ is a subset of a conjugate of $P$.

But since $\order P = \order Q$, it follows that $Q$ must equal a conjugate of $P$.

$\blacksquare$


Proof 2

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

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 left cosets of $G \pmod H$.

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

$g * S_i = g S_i$.

Let $H_i$ denote the stabilizer of $S_i$.


By the Orbit-Stabilizer Theorem:

$\order {H_i} = p^n$

while:

$S_i = g H \implies g H g^{-1} \subseteq H_i$

Because $\order {g H g^{-1} } = \order H = \order {H_i}$, we have:

$g H g^{-1} \subseteq H_i$


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

Then $H'$ acts on $G / H$ by the same rule as $G$.

Since $p \nmid m$, there exists at least one orbit under $H'$ whose cardinality is not divisible by $p$.

Suppose that $S_1, S_2, \ldots, S_r$ are the elements of an orbit where $p \nmid r$.

Let $K = H' \cap H_1$.

Then $K$ is the stabilizer of $S_1$ under the action of $H'$.

Therefore:

$\index {H'} K = r$

However:

$\order {H'} = p^n$

and:

$p \nmid r$

from which it follows that:

$r = 1$

and:

$K = H'$


Therefore:

$\order K = \order {H'} = \order {H_1} = p^n$

and:

$H' = K = H_1$

Thus $H'$ and $H$ are conjugates.


Also known as

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

Others call this the second 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.