Sylow Theorems

When cracking open the structure of a group, it is a useful plan to start with investigating the subgroups of prime order. 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. The name is pronounced something like "Soolof".

There is no standard numbering for the Sylow Theorems. Different authors use different labellings. Therefore, the following nomenclature is to a greater or lesser extent arbitrary.

First Sylow Theorem

Let $p$ be a prime number, and let $G$ be a group such that $\left\vert{G}\right\vert = k p^n$ where $p \nmid k$.

Then $G$ has at least one Sylow $p$-subgroup.

Second Sylow Theorem

Let $P$ be a Sylow $p$-subgroup of the finite group $G$.

Let $Q$ be any $p$-subgroup of $G$.

Then $Q$ is contained in a conjugate of $P$.

Some sources call this the third Sylow theorem.

Third Sylow Theorem

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

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.

Fourth Sylow Theorem

The number of Sylow $p$-subgroups of a finite group is congruent to $1 \left({\bmod p}\right)$.

Some sources call this the second Sylow theorem.

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

Fifth Sylow Theorem

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

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.