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

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?