# Sylow Theorems

$$\newcommand{\ZZ}{\Bbb Z} \DeclareMathOperator{\Stab}{Stab} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\sgn}{sgn}$$

In group theory, the Sylow theorems are a triplet of theorems that pin down a suprising amount of information about certain subgroups.

Lagrange’s theorem tells us that if $$H$$ is a subgroup of $$G$$, then the size of $$H$$ divides the size of $$G$$. The Sylow theorems give us some answers to the converse question: for what divisors of $$|G|$$ can we find a subgroup of that size?