Key Concepts of Sylow Theorems

Why This Matters

The Sylow Theorems are among the most powerful tools in finite group theory, giving you a systematic way to decompose and analyze groups based on their prime factorizations. You're being tested on your ability to use these theorems to determine subgroup existence, count possible configurations, and identify when subgroups must be normal—skills essential for group classification, structure analysis, and proving properties about specific groups.

These theorems connect directly to fundamental concepts like Lagrange's Theorem, conjugacy classes, normal subgroups, and group actions. When you encounter a group of order $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, the Sylow Theorems tell you exactly what subgroups must exist and constrain how many there can be. Don't just memorize the theorem statements—know how to apply each one to determine group structure and recognize when a Sylow subgroup must be normal.

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The Three Sylow Theorems

These three results work together as a complete toolkit: the first guarantees existence, the second establishes structural uniformity through conjugacy, and the third constrains the count. Mastering when and how to apply each theorem is the key to solving classification problems.

First Sylow Theorem (Existence)

• Guarantees existence of Sylow p-subgroups—for any prime $p$ dividing $|G|$, at least one Sylow $p$-subgroup exists in $G$
• Specifies the subgroup order precisely: if $|G| = p^n m$ where $p \nmid m$, then a Sylow $p$-subgroup has order exactly $p^n$
• Foundation for all structure analysis—without guaranteed existence, the other theorems would have nothing to work with

Second Sylow Theorem (Conjugacy)

• All Sylow p-subgroups are conjugate—if $P$ and $Q$ are Sylow $p$-subgroups, then $gPg^{-1} = Q$ for some $g \in G$
• Implies structural equivalence: conjugate subgroups are isomorphic, so all Sylow $p$-subgroups share identical algebraic properties
• Key for normality arguments—a Sylow $p$-subgroup is normal if and only if it's the unique one (since conjugation would map it to itself)

Third Sylow Theorem (Counting)

• Constrains the number $n_p$ of Sylow p-subgroups with two conditions: $n_p \mid |G|$ and $n_p \equiv 1 \pmod{p}$
• Often forces $n_p = 1$: when the only divisor of $|G|$ congruent to 1 mod $p$ is 1 itself, the Sylow $p$-subgroup must be unique (hence normal)
• Primary tool for group classification—these arithmetic constraints frequently eliminate possible group structures

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Foundational Definitions

Understanding what Sylow $p$-subgroups actually are is essential before applying the theorems. The definition encodes both the prime power structure and the maximality condition.

Definition of a Sylow p-subgroup

• Maximal p-subgroup of G—a subgroup of order $p^n$ where $p^n$ is the largest power of $p$ dividing $|G|$
• Not contained in any larger p-subgroup: the maximality condition distinguishes Sylow $p$-subgroups from arbitrary $p$-subgroups
• Captures all "p-power structure" of the group—essential for decomposing $G$ by its prime factors

Connection to Lagrange's Theorem

• Lagrange provides the foundation—subgroup orders must divide group order, which the Sylow Theorems refine for prime powers
• Sylow Theorems are a partial converse: while Lagrange doesn't guarantee subgroups of every divisor order exist, Sylow guarantees them for maximal prime powers
• Together they bound subgroup structure—Lagrange constrains what's possible, Sylow confirms what's guaranteed

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Normality and Uniqueness

The connection between Sylow subgroups and normal subgroups is one of the most frequently tested applications. When $n_p = 1$, the unique Sylow $p$-subgroup must be normal, which often simplifies group structure dramatically.

Sylow p-subgroups and Normal Subgroups

• Unique Sylow p-subgroup implies normality—if $n_p = 1$, then the single Sylow $p$-subgroup $P$ satisfies $gPg^{-1} = P$ for all $g \in G$
• Conjugacy (Second Theorem) explains why: since all Sylow $p$-subgroups are conjugate, having only one means it must be fixed under conjugation
• Normal Sylow subgroups enable quotient constructions—finding normal subgroups is often the first step in analyzing group structure via quotients

Counting Arguments for Normality

• Apply Third Sylow Theorem constraints—list divisors of $|G|$ that are $\equiv 1 \pmod{p}$ to find possible values of $n_p$
• Intersection of conditions often forces $n_p = 1$: when $n_p \mid m$ (where $|G| = p^n m$) and $n_p \equiv 1 \pmod{p}$ leave only one option
• Standard technique in classification problems—this arithmetic argument appears constantly in proving groups of certain orders have normal subgroups

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Applications in Group Classification

The Sylow Theorems transform abstract group classification into concrete arithmetic. By analyzing the prime factorization of $|G|$, you can often determine the group's structure completely.

Classification Strategy

• Factor the group order as $|G| = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ and apply Sylow Theorems to each prime
• Determine $n_{p_i}$ constraints for each prime—often multiple primes yield $n_p = 1$, giving multiple normal subgroups
• Use normal subgroups to decompose G—groups with normal Sylow subgroups for all primes are often direct products

Identifying Simple Groups

• Simple groups have no proper normal subgroups—so no Sylow subgroup can be unique (except in trivial cases)
• Sylow counting eliminates candidates: if any $n_p = 1$ is forced, the group cannot be simple
• Foundational for classification of finite simple groups—Sylow analysis is the first step in most simplicity proofs

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Quick Reference Table

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Self-Check Questions

1. If $|G| = 56 = 2^3 \cdot 7$, what are the possible values of $n_7$? What does each value tell you about the group's structure?

2. Explain why the Second Sylow Theorem implies that if $n_p = 1$, the unique Sylow $p$-subgroup must be normal in $G$.

3. Compare and contrast what Lagrange's Theorem and the First Sylow Theorem tell you about subgroups of a group of order 12.

4. A group $G$ has order $p^2 q$ where $p < q$ are primes and $q \not\equiv 1 \pmod{p}$. Which Sylow subgroup(s) must be normal, and why?

5. How would you use the Sylow Theorems to begin proving that every group of order 15 is cyclic?