Sylow's First Theorem

Sylow’s First Theorem. Let $G$ be a finite group , and write $|G| = p^{a}m$ where $p$ is prime and $p \nmid m$. Then $G$ has a subgroup of order $p^{a}$. Any subgroup of order $p^{a}$ is called a Sylow p-subgroup .

Sylow’s first theorem vastly strengthens Cauchy's theorem (the case $a=1$). Together with Sylow's second theorem and Sylow's third theorem , it controls the existence and placement of maximal p-groups inside $G$.

Proof sketch. A standard proof uses a carefully chosen group action on a set whose size is divisible by $p$ (often involving subsets or tuples) and then applies counting/orbit arguments to force a stabilizer of size $p^a$. The stabilizer is the desired subgroup of order $p^a$.