Sylow theorems

Remarks

Each part has its own node: the first Sylow theorem, the second Sylow theorem and the third Sylow theorem. The rest of part 3 is in the nodes $n_p$ divides the index and $n_p$ is the index of the normaliser. The usual proofs let a group act on a set and count orbits or fixed points. Part 1 uses the action on subsets of size $p^n$, or induction on $\abs{G}$. Parts 2 and 3 let a $p$-group act on conjugates or cosets, and count its fixed points modulo $p$.

Example: every group $G$ of order $15 = 3 \cdot 5$ is cyclic. By part 3, $n_3$ divides $5$ and $n_3 \equiv 1 \pmod 3$, so $n_3 = 1$; and $n_5$ divides $3$ and $n_5 \equiv 1 \pmod 5$, so $n_5 = 1$. So $G$ has a normal subgroup of order $3$ and one of order $5$. Their intersection is $\set{e}$, since its order divides both $3$ and $5$. Hence $G \cong \Z/3\Z \times \Z/5\Z \cong \Z/15\Z$.

The prime power in part 1 matters: the converse of Lagrange's theorem fails for other divisors, since $A_4$ has no subgroup of order $6$.

Going deeper

For a student

The Sylow theorems are the first main tool to study finite groups through their order alone. A typical use: the counting rules in part 3 force $n_p = 1$, so the Sylow subgroup is normal and the group is not simple. In this way one shows, for example, that no group of order $20$ and no group of order $pq$, with primes $p < q$, is simple. The theorems lead to the Frattini argument, to Hall subgroups in solvable groups, and to the study of how $p$-elements are conjugate, as in the focal subgroup theorem.