Sylow Theorems: Statements, Proofs Sketch & Applications

The Sylow theorems describe the p-subgroups of a finite group. Write |G| = p^a · m with p ∤ m; a Sylow p-subgroup has order p^a, the largest p-power dividing |G|. The three theorems guarantee these subgroups exist, show they are all conjugate, and tightly restrict how many there can be.

They are the main tool for proving groups of a given order must have certain structure — e.g. every group of order 15 is cyclic, and no simple group has order 30. Sylow theory turns the prime factorization of |G| into hard facts about subgroups.

Key Concepts

• Sylow p-subgroup: subgroup of order p^a, the full p-part of |G|.
• First theorem: Sylow p-subgroups exist for every prime p | |G|; every p-subgroup sits inside one.
• Second theorem: all Sylow p-subgroups are conjugate in G.
• Third theorem: n_p ≡ 1 (mod p) and n_p | m (n_p = number of Sylow p-subgroups).
• Normality test: n_p = 1 iff the Sylow p-subgroup is normal in G.

Formulas

• Restrictions: n_p ≡ 1 (mod p) and n_p | m, where |G| = p^a · m. (Pure counting; no physical units.)
• Element counting: distinct Sylow p-subgroups intersect in fewer than p^a elements, so counts can force contradictions.
• Limit: Sylow theory needs finite G; it restricts which n_p are possible, not which occurs.

Worked Example

Take G = S_3, |G| = 6 = 2 · 3. For p = 3: n_3 ≡ 1 (mod 3) and n_3 | 2, so n_3 = 1 — the unique Sylow 3-subgroup {e, (123), (132)} is normal. For p = 2: n_2 ≡ 1 (mod 2) and n_2 | 3, so n_2 ∈ {1, 3}; the order-2 subgroups are {e,(12)}, {e,(13)}, {e,(23)}, so n_2 = 3, none normal. Pure arithmetic forces the 3-subgroup to be normal while the 2-subgroups need not be.

Common Misconceptions

• "n_p = 1 always." Not so — in S_3, n_2 = 3. Uniqueness forces normality, but Sylow subgroups are frequently non-unique.
• "Every divisor of |G| occurs as a subgroup order." False in general (A_4 has order 12 but no subgroup of order 6); Sylow only guarantees p-power orders.
• "Sylow subgroups for different primes are unrelated." Counting elements across Sylow subgroups for all primes simultaneously is exactly how simplicity proofs work — the subgroups constrain each other through the total element count.

Real-World Applications

• Group classification: proving groups of orders like 15, 21, 35, or pq are cyclic or have normal subgroups.
• Simplicity tests: showing no simple group exists of many orders (for example 30, 36, 48) by forcing a normal Sylow subgroup.
• Galois theory: Sylow subgroups of Galois groups correspond to intermediate fields with restricted ramification behavior.
• Representation theory: the structure of p-subgroups controls modular representations in characteristic p.

The Sylow theorems squeeze the possible subgroup structure of a finite group out of the prime factorization of its order — existence, conjugacy, and tight counting restrictions, all from one factorization.