p-groups

A p-group is a finite group whose order is a power of a prime p.

Structure

Fix a prime p. A p-group has order pᵏ for some k ≥ 0.

• Nilpotent. Every p-group is nilpotent, hence supersolvable and solvable. It is its own Sylow p-subgroup.
• Nontrivial center. Every nontrivial p-group has a nontrivial center, Z(G) ≠ {e}. (Class equation: each non-central class has size divisible by p, so |Z(G)| ≡ |G| ≡ 0 (mod p).) More strongly, every nontrivial normal subgroup meets the center: N ∩ Z(G) ≠ {e}.
• Normal subgroup of every order. A group of order pᵏ has a normal subgroup of order pʲ for every 0 ≤ j ≤ k, and a chief series whose factors are all Cₚ (each chief factor is central of order p). In particular every maximal subgroup has index p, is normal, and contains the Frattini subgroup.
• Generators — Burnside basis theorem. The Frattini subgroup is Φ(G) = Gp[G, G] (see Frattini, commutator); the quotient G/Φ(G) is elementary abelian, and its dimension over the field 𝔽ₚ of p elements equals the minimum number of generators of G.
• Order p² is abelian. Every group of order p² is abelian: either the cyclic Cp² or the elementary abelian Cₚ × Cₚ.
• Order p³ — exactly five. For every prime p there are exactly 5 groups of order p³: three abelian (Cp³, Cp² × Cₚ, Cₚ × Cₚ × Cₚ) and two non-abelian, both extraspecial. For p = 2 the non-abelian pair is D₄ and Q₈.
• Counting. The number of groups of order pⁿ grows very fast in n. For p = 2 the totals at orders 2, 4, 8, 16, 32, 64 are 1, 2, 5, 14, 51, 267; asymptotically the number of groups of order pⁿ is p(2/27)n³ + O(n8/3). Small groups are overwhelmingly p-groups: over 99% of the groups of order at most 2000 have order 1024.

Examples

Abelian families.

• Cyclic Cₚⁿ — the only cyclic p-groups: C₄, C₈, C₉, …
• Elementary abelian Cₚⁿ = Cₚ × ⋯ × Cₚ — exponent p: the Klein four-group C₂×C₂, C₃×C₃, …

Non-abelian families — each an infinite series of p-groups:

• Extraspecial p(1+2n)₊/₋, order p2n+1 — center, commutator, and Frattini subgroup all equal Cₚ; two ATLAS types ₊/₋ for each p and n. Smallest are the order-8 pair dihedral D₄ and quaternion Q₈.
• Heisenberg UT(3,𝔽p), order p³ — the 3 × 3 upper-unitriangular matrices over 𝔽ₚ. It is exactly the smallest ₊-type extraspecial group p(1+2)₊; for odd p it has exponent p (every non-identity element has order p), while UT(3,𝔽2) ≅ D₄ has exponent 4. Its order-p³ partner p(1+2)₋ (exponent p²) is extraspecial but not Heisenberg. For p = 3: 3(1+2)₊.
• Generalized quaternion Q₂ⁿ, order 2ⁿ (n ≥ 3) — maximal class, with a single involution: Q₈, Q₁₆, Q₃₂, … Its maximal-class siblings are the dihedral and semi-dihedral 2-groups.
• Modular maximal-cyclic M₂ⁿ, order 2ⁿ (n ≥ 4) — a cyclic subgroup of index 2 but only nilpotency class 2: M₁₆, M₃₂, …

Properties

• Every non-trivial finite p-group has a non-trivial center.
• Every finite p-group is nilpotent.
• Every finite nilpotent group is the direct product of its Sylow subgroups.
• A finite group is nilpotent exactly when it is the direct product of its Sylow subgroups: G ≅ ∏p Gp, one factor per prime.

See also

• Nilpotent group — every p-group is nilpotent; a finite nilpotent group is the direct product of its Sylow p-subgroups.
• Center — nontrivial for every nontrivial p-group.
• Frattini subgroup — Φ(G) = Gp[G, G]; G/Φ(G) fixes the minimum generating set.
• Extraspecial group — the non-abelian groups of order p³ and their higher analogs.
• Generalized quaternion, dihedral — the maximal-class families of order 2ⁿ.
• Sylow subgroup — a p-group is its own Sylow p-subgroup.
• Wikipedia: p-group