Finite cyclic group
A group with finitely many elements that can all be generated by powers of a single element. Its order equals the order of any generator.
Group (mathematics): A set equipped with an associative operation, an identity element, and an inverse for every element. Finite cyclic groups satisfy the group axioms under their defining operation.
Cayley graph: A graph encoding a group's multiplication by connecting elements through selected generators. With one cyclic generator, the graph traces a cycle through all group elements.
Clock arithmetic: Arithmetic on a repeating cycle of values, commonly represented by integers modulo a fixed number. A clock's hour positions form a cyclic group under addition modulo its period.
Infinite cyclic group: A group isomorphic to the integers under addition, generated by one element of infinite order. It shares single-generation with finite cyclic groups but has infinitely many elements.
Cyclic group: A group generated by one element, so every element is a power of that generator. Finite cyclic groups are precisely the cyclic groups whose underlying sets are finite.
Subgroup: A subset of a group that is itself a group under the same operation. Every subgroup of a finite cyclic group is cyclic, with one subgroup for each divisor of its order.
Roots of unity: Complex numbers whose positive integer powers equal one. The nth roots of unity form a finite cyclic group under multiplication.
Symmetric group: A group of permutations of a set, with composition as its operation. Symmetric groups of degree at least three are generally noncyclic, unlike every finite cyclic group.
Group order: The number of elements in a finite group. For a finite cyclic group, this number also determines its isomorphism type.
Euler's totient function: The function φ(n) counts the positive integers up to n that are relatively prime to n. A cyclic group of order n has exactly φ(n) generators.