State the three Sylow theorems.
Solution
Let \(G\) be a finite group and let \[ |G|=p^n m, \qquad p\nmid m. \] A subgroup of order \(p^n\) is called a Sylow \(p\)-subgroup.
The Sylow theorems are:
1. Existence.There exists a subgroup\[ P\le G \]of order \(p^n\).
2. Conjugacy and containment.Every \(p\)-subgroup of \(G\) is contained in a Sylow \(p\)-subgroup, and any two Sylow \(p\)-subgroups are conjugate in \(G\).
3. Number of Sylow subgroups.If \(n_p\) denotes the number of Sylow \(p\)-subgroups, then\[ n_p\equiv1\pmod p \]and\[ n_p\mid m. \]
In particular, \[ n_p=1 \iff \text{the Sylow $p$-subgroup is normal in $G$}. \]