# Sylow Theorems/Historical Note

## Historical Note on Sylow Theorems

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.

### First Sylow Theorem

Sylow's original work in $1872$ demonstrated the existence of what is now known as a Sylow $p$-subgroup.

The corollary, that there exists a subgroup of order $p^n$ for all $p^n \divides \order G$, was deduced later, but is frequently itself referred to as the First Sylow Theorem.

The proof using the Orbit-Stabilizer Theorem is based on one published by Helmut Wielandt in $1959$.

## Sources

- 1872: Peter Ludwig Mejdell Sylow:
*Théorèmes sur les Groupes de Substitutions*(*Math. Ann.***Vol. 5**: pp. 584 – 594) - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 44$. Some consequences of Lagrange's Theorem - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $11$: The Sylow Theorems: Summary for Chapter $11$