procyclic

Procyclic refers to a property of certain profinite groups. A profinite group G is procyclic if it can be generated by a single element in the topological sense: there exists an element g in G whose closed cyclic subgroup ⟨g⟩ is dense in G, and hence equals G. Equivalently, every finite quotient of G is cyclic, and G can be expressed as the inverse limit of finite cyclic groups with surjective bonding maps: G ≅ lim← C_n_i.

Consequences and structure: Procyclic groups are compact, Hausdorff, and totally disconnected. They are precisely the profinite groups whose finite quotients are all cyclic. This makes them the inverse limits of finite cyclic groups, and they can be described as closed subgroups or quotients of the profinite completion of the integers. A standard example is the profinite completion of Z, denoted by Ẑ, which is itself procyclic and isomorphic to the product ∏_p Z_p. Any procyclic group is a quotient of Ẑ by a closed subgroup, and any closed subgroup of a procyclic group is itself procyclic.

Examples and non-examples: Finite cyclic groups C_n are procyclic (they are already finite and generated by one element). The absolute Galois group of a finite field, Gal(F̄_q / F_q), is procyclic and generated by the Frobenius automorphism; it is isomorphic to Ẑ. The full product ∏_p Z_p, while procyclic as a limit, is not a finite quotient and serves as a canonical model in the profinite setting. A product of two distinct p-adic factors, such as Z_p × Z_q with distinct primes p ≠ q, is not procyclic because it has noncyclic finite quotients.

Applications and context: Procyclic groups arise naturally in number theory and algebraic geometry, particularly in Galois theory and the study of profinite fundamental groups. They provide a simplifying case where the entire group is controlled by a single topological generator.