Sylow's Theorems

Sylow Subgroups

If $|G| = p^m \cdot k$, where $p$ is prime and $p \nmid k$, then a subgroup of order $p^m$ is called a Sylow p-subgroup.

The Three Sylow Theorems (1872):

1. Existence: In a finite group $G$ there exists a Sylow $p$-subgroup for every prime $p$.

2. Conjugacy: All Sylow $p$-subgroups are conjugate: if $P$ and $P'$ are Sylow $p$-subgroups, then $P' = gPg^{-1}$ for some $g$.

3. Number: The number $n_p$ of Sylow $p$-subgroups satisfies: $n_p \equiv 1 \pmod{p}$ and $n_p \mid |G|/p^m$.

Applications of Sylow's Theorems

Groups of order $15 = 3 \cdot 5$: $n_3 \mid 5$ and $n_3 \equiv 1 \pmod{3} \implies n_3 = 1$. $n_5 \mid 3$ and $n_5 \equiv 1 \pmod{5} \implies n_5 = 1$. Both Sylow subgroups are normal $\rightarrow G \cong \mathbb{Z}_{15}$ (the unique group of order $15$).

Groups of order $pq$ ($p < q$, $p \nmid q-1$): likewise $G \cong \mathbb{Z}_{pq}$.

Nilpotent and Solvable Groups

A group is nilpotent if the descending central series $G \supseteq G' \supseteq G'' \supseteq \ldots$ reaches ${e}$. A finite group is nilpotent $\iff$ it is the direct product of its Sylow subgroups.

A group is solvable if there is a series $G = G_0 \supseteq G_1 \supseteq \ldots \supseteq G_k = {e}$ with abelian factors $G_i/G_{i+1}$.

Galois Theorem: The equation $pn(x) = 0$ is solvable in radicals $\iff$ its Galois group is solvable. The Galois group of a general equation of degree $\geq 5$ is $S_n$, which is not solvable for $n \geq 5$.