Cyclic number (group theory)¶

A positive integer n such that every group of order n is cyclic, equivalently n is coprime to Euler's totient phi(n).

Version: v1 · 2026-09-08 · History
Domain-specific #: 4012
Origin domain: finite group theory and number theory
Subdomain: finite group theory and number theory

Core Idea¶

Cyclic numbers are square-free and satisfy a prime-divisibility criterion preventing any prime factor of n from dividing another factor minus one. Prime factorization controls the possible Sylow actions and semidirect products; the coprimality condition removes noncyclic extensions, forcing every group of that order to be cyclic. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of finite group theory and number theory. It is the domain-specific identity determined by the integer is positive, Euler-totient and group-order conventions are explicit, gcd(n,phi(n)) equals one, and the equivalent square-free prime-divisibility conditions and all-groups conclusion are stated.

Scope of Application¶

Cyclic number (group theory) belongs to finite group theory and number theory and is useful where the analyst can specify the typed finite group theory and number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the integer is positive, Euler-totient and group-order conventions are explicit, gcd(n,phi(n)) equals one, and the equivalent square-free prime-divisibility conditions and all-groups conclusion are stated. The scope is broad within that domain but bounded by the need for the integer is positive, Euler-totient and group-order conventions are explicit, gcd(n,phi(n)) equals one, and the equivalent square-free prime-divisibility conditions and all-groups conclusion are stated.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the integer is positive, Euler-totient and group-order conventions are explicit, gcd(n,phi(n)) equals one, and the equivalent square-free prime-divisibility conditions and all-groups conclusion are stated the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Cyclic number (group theory). Cyclic number (group theory) compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed finite group theory and number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the integer is positive, Euler-totient and group-order conventions are explicit, gcd(n,phi(n)) equals one, and the equivalent square-free prime-divisibility conditions and all-groups conclusion are stated independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of finite group theory and number theory because they reuse the typed finite group theory and number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Prime factorization controls the possible Sylow actions and semidirect products; the coprimality condition removes noncyclic extensions, forcing every group of that order to be cyclic., and type the carrier, state every parameter and convention in the definition, test that the integer is positive, Euler-totient and group-order conventions are explicit, gcd(n,phi(n)) equals one, and the equivalent square-free prime-divisibility conditions and all-groups conclusion are stated, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Cyclic number (group theory) Domain-specific

Parents (1) — more general patterns this builds on

Cyclic number (group theory) is a kind of Classification Prime

The proposed strict upward parent is prime:classification.

Hierarchy path (1) — routes to 1 parentless root

Cyclic number (group theory) → Classification

Neighborhood in Abstraction Space¶

Cyclic number (group theory) sits in a crowded region of the domain-specific corpus (16^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Group & Semigroup Structure (26 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08