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 $11$