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 $$\left|{G}\right| = k p^n$$ where $$p \nmid k$$.

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

See First Sylow Theorem.

Second Sylow Theorem
If $$P$$ is a Sylow $p$-subgroup of the finite group $$G$$, and $$Q$$ is any $p$-subgroup of $$G$$, then $$Q$$ is contained in a conjugate of $$P$$.

See Second Sylow Theorem.

Some sources call this the third Sylow theorem.

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

See Third Sylow Theorem.

Some sources call this the fourth Sylow theorem, and merges 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 $$1 \left({\bmod\, p}\right)$$.

See Fourth Sylow Theorem.

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.

See Fifth Sylow Theorem.

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.