Sylow’s Theorems

Key Definitions

Definition.

  1. A group $G$ is called a $p$–group if every element of $G$ has order $p^k$ for some non-negative integer $k$. A finite group $G$ is a $p$–group if and only if $|G| = p^n$ for some $n$.

  2. Let $G$ be a finite group such that $p^n \mid |G|$ but $p^{n+1} \nmid |G|$. A Sylow $p$–subgroup of $G$ is a subgroup of order $p^n$. We denote the set of Sylow $p$–subgroups of $G$ by $\mathrm{Syl}_p(G)$.

Key Theorems

Theorem (Sylow’s First Theorem).
Let $G$ be a finite group with $p^n \mid |G|$. Then $G$ contains a subgroup of order $p^n$. In fact, there exist subgroups

\[P_1 \trianglelefteq P_2 \trianglelefteq \cdots \trianglelefteq P_n \leq G\]

such that $|P_i| = p^i$ for each $i$.

Theorem (Sylow’s Second Theorem).
Let $G$ be a finite group, $Q \leq G$ a $p$–group, and $P \in \mathrm{Syl}_p(G)$. Then there exists $g\in G$ such that

\[gQg^{-1} \leq P.\]

As a corollary:

  • $G$ acts transitively on $\mathrm{Syl}_p(G)$ by conjugation.

  • For $P \in \mathrm{Syl}_p(G)$,

    \[|\mathrm{Syl}_p(G)| = [G : N_G(P)].\]

Theorem (Sylow’s Third Theorem).
Let $G$ be a finite group and $p$ a prime dividing $|G|$. Then

\[|\mathrm{Syl}_p(G)| \equiv 1 \pmod{p}\]

Exercises

Easy:

  1. Let $G$ be a group of order $pq$, where $p$ and $q$ are primes with $p < q$. Show that $G$ has a normal subgroup of order $q$.
  2. Let $G$ be a group of order $p(p-1)$ where $p$ is prime. Show that $G$ has a subgroup of order $p$.
  3. (UCI 2025 June Algebra Qual) Let $G$ be a finite group and $p$ a prime dividing $|G|$. Let $O_p(G)$ denote the intersection of all Sylow $p$–subgroups of $G$. Show that $O_p(G) \trianglelefteq G$ and that it contains every normal $p$–subgroup of $G$.

Medium:

  1. (UCI 2024 September Algebra Qual) Let $G$ be a group of order $p^k (p+1)$ where $p$ is prime and $k \geq 2$. Show that $G$ is not simple.
    (Hint: Show that $s_p = |\mathrm{Syl}_p(G)|$ is either $1$ or $p+1$. In the latter case, pick $P \in \mathrm{Syl}_p(G)$ and consider the induced homomorphism

    \[\rho: G \to S_{G/P} \cong S_{p+1}.\]

    Analyze $\ker \rho$.)
    (Hint: You can also use the Burnside’s normal p-complement theorem to overkill this problem. )