pGruppen

p-Gruppen are finite groups whose order is a power of a prime p. They are a central object in finite group theory and play a key role in Sylow theory and the local analysis of larger groups.

Key properties include that every finite p-group is nilpotent and has a nontrivial center. The upper central series reaches the whole group, and the Frattini subgroup Φ(G)—the intersection of all maximal subgroups—controls generating sets: for a finite p-group G, the minimal number of generators d(G) equals the dimension of G/Φ(G) as a vector space over the field with p elements. Subgroups of index p are normal, since the quotient has order p.

Examples range from the cyclic group of order p^n to the elementary abelian p-group (C_p)^n. Non-abelian p-groups include extra-special p-groups of order p^3. These examples illustrate the diversity of p-groups even at small orders.

Classification of p-groups by order p^n is feasible only for small n; beyond that, there is no complete general classification. The landscape grows rapidly with n, and coclass theory provides a framework to organize families of p-groups with fixed p and varying order.

Pro-p groups generalize finite p-groups by taking inverse limits of finite p-groups; they carry a natural topology and arise in number theory and algebra, notably as Galois groups of p-extensions and in the study of p-adic analytic groups.

See also: Sylow theorems, nilpotent groups, Frattini subgroup, Burnside basis theorem, pro-p groups.