Sylow p-Subgroup is Unique iff Normal

Theorem
A group $$G$$ has exactly one Sylow $p$-subgroup $$P$$ iff $$P$$ is normal.

Proof

 * If $$G$$ has precisely one Sylow $p$-subgroup, it must be normal from Unique Subgroup of a Given Order is Normal.


 * Suppose a Sylow $p$-subgroup $$P$$ is normal. Then it equals its conjugates.

Thus, by the Third Sylow Theorem, there can be only one such Sylow $p$-subgroup.