Sylow Theorems

A Sylow p-subgroup of a finite group G is a subgroup whose order is the highest power of the prime p dividing |G|. Write |G| = pᵏ·m with p not dividing m. Then Sylow p-subgroups have order pᵏ: in a group of order 12, that is order 4 for p = 2 and order 3 for p = 3 (Conrad).

Here n_p is the number of Sylow p-subgroups of G. Two subgroups P and Q are conjugate when Q = gPg⁻¹ for some g in G, and a subgroup is normal when it equals all its conjugates. The statement n_p ≡ 1 (mod p) means n_p leaves remainder 1 on division by p.

• Sylow I (existence): a Sylow p-subgroup exists for every prime p, and every subgroup of p-power order lies inside one.
• Sylow II (conjugacy): any two Sylow p-subgroups are conjugate.
• Sylow III (counting): n_p divides m and n_p ≡ 1 (mod p).
• Sylow III*: n_p equals the index of the normaliser N(P), the set of g with gPg⁻¹ = P.

A normal subgroup is its own only conjugate, so by Sylow II a Sylow p-subgroup is normal exactly when n_p = 1.

A₄ has one subgroup of order 4 and four of order 3, while the dihedral group of order 12 has three of order 4 and one of order 3 (Conrad). The theorems restrict the counts. They do not decide them.

List the divisors of m = |G|/pᵏ and keep those that leave remainder 1 when divided by p. What survives are the possible values of n_p.

Finding possible n_p, step by step

1. Factor |G| = pᵏ·m with p not dividing m.
2. Write every divisor of m.
3. Cross out each divisor that is not ≡ 1 (mod p).
4. If only 1 remains, the Sylow p-subgroup is unique, hence normal, and G (with order not a prime power) is not simple. A simple group has no normal subgroups except the trivial one and itself.

For |G| = 12 with p = 2: m = 3, divisors 1 and 3, both odd, so n₂ ∈ {1, 3}. With p = 3: m = 4, divisors 1, 2, 4, and 2 fails, so n₃ ∈ {1, 4}.

For 117 of the 139 orders from 2 to 200 that are not prime powers (84%), the table forces a normal Sylow subgroup. This count, run in October 2026, covers every order n from 2 to 200 that is not a prime power (for a prime power the Sylow subgroup would be the whole group, as Wikipedia notes).

The computation uses the Sylow III conditions (n_p divides m, n_p ≡ 1 mod p) to build the allowed n_p list for each prime dividing n, then two tests:

• Level 1: some prime allows only n_p = 1.
• Level 2: a simple non-abelian group of order n with n_p > 1 needs n to divide n_p!/2. If, for some prime, no allowed n_p > 1 passes, the order is excluded as non-simple. This refines the coset-action argument in Math3ma.

Orders excluded at Level 2 up to 200: 12, 24, 36, 48, 72, 80, 96, 108, 112, 150, 160 and 192. The survivors resist both tests and need a case-specific argument.

The counts say which orders can be finished by table. They say nothing about which groups exist. The table below shows common exam orders.

Use a second argument: count elements, let the group act on its Sylow subgroups, or compare the intersection of two Sylow subgroups.

The costliest is forcing the maximum counts. For order 42, taking n₂ = 21 and n₃ = 7 gives 6 + 21 + 14 + 1 = 42 elements, so it seems both are forced. That is false: Z/42 has n₂ = n₃ = 1 and Aff(Z/7) has n₂ = n₃ = 7, because some elements have other orders (Conrad). Use element counts only to rule combinations out when they exceed |G|.

• Reading the table as the answer. The table gives candidates only; decide the actual n_p from the specific group.
• Conjugate subgroups versus conjugate elements. In A₄, ⟨(123)⟩ and ⟨(124)⟩ are conjugate, but (123) and (124) are not; conjugate (123) to (142) instead (Conrad).
• Over-extending Sylow II. In a group of order 100, the subgroups of order 4 and of order 25 are conjugate; those of order 2 or 5 need not be (Conrad).

• Lagrange's theorem: subgroup order divides group order. Sylow I is a partial converse for prime powers.
• p-group: a group whose order is a power of p. Every Sylow subgroup is one.
• Normaliser N(P): the elements that conjugate P to itself. Its index is n_p.
• Cayley's theorem: every group embeds in a permutation group, the idea behind the action arguments above.

The GATE and CSIR-NET syllabi name them; the Common University Entrance Test for Postgraduate admissions (CUET PG) syllabus does not. The GATE 2027 Mathematics syllabus lists "Sylow's theorems and their applications" in Section 6, Algebra (Indian Institute of Technology Madras).

The CSIR-UGC NET Mathematical Sciences syllabus on the Council of Scientific and Industrial Research Human Resource Development Group (CSIR HRDG) site lists "Sylow theorems" in its algebra topics, as of October 2026 (CSIR HRDG).

The CUET PG Mathematics (SCQP19) syllabus does not name Sylow theorems; its group theory stops at Lagrange's theorem, homomorphisms and quotient groups (National Testing Agency, NTA).

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