Sylow's Second Theorem

Sylow’s Second Theorem. Let $G$ be a finite group , let $p$ be a prime, and let $P$ be a Sylow p-subgroup of $G$. If $Q \le G$ is any p-group (equivalently, $|Q|$ is a power of $p$), then there exists $g\in G$ such that

In particular, any two Sylow $p$-subgroups of $G$ are conjugate (they lie in the same orbit under the conjugation action ).

Sylow’s second theorem implies Sylow $p$-subgroups are “unique up to conjugacy,” and it is the key input for the normality test n_p=1 implies the Sylow p-subgroup is normal . It is typically proved using Sylow's first theorem plus an action on cosets.

Proof sketch. Let $Q$ act by left multiplication on the set of left cosets $G/P$. Orbit sizes are powers of $p$, so the number of fixed points is congruent to $|G/P|$ modulo $p$. A fixed point corresponds to a coset $gP$ with $Q \subseteq gPg^{-1}$, forcing such a conjugate to contain $Q$.