Sylow Theorems
Theorem
First Sylow Theorem
Let $p$ be a prime number.
Let $G$ be a group such that:
- $\order G = k p^n$
where:
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 a subset of a conjugate of $P$.
Third Sylow Theorem
All the Sylow $p$-subgroups of a finite group are conjugate.
Fourth Sylow Theorem
The number of Sylow $p$-subgroups of a finite group is congruent to $1 \pmod p$.
Fifth Sylow Theorem
The number of Sylow $p$-subgroups of a finite group is a divisor of their common index.
Examples
Sylow $2$-Subgroups in Group of Order $12$
In a group of order $12$, there are either $1$ or $3$ Sylow $2$-subgroups.
Sylow $3$-Subgroups in Group of Order $12$
In a group of order $12$, there are either $1$ or $4$ Sylow $3$-subgroups.
Sylow $7$-Subgroups in Group of Order $28$
In a group of order $28$, there exists exactly $1$ Sylow $7$-subgroup.
This Sylow $7$-subgroup is normal in $G$.
Sylow $2$-Subgroups in Group of Order $32$
The Sylow Theorems tell us nothing about the Sylow $2$-subgroups in a group of order $32$.
Sylow $2$-Subgroups in Group of Order $48$
In a group of order $48$, there are either $1$ or $3$ Sylow $2$-subgroups.
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.