Knowra Cyclic group Cyclic group A group generated by one element, whose integer powers—or multiples in additive notation—produce every element. Every cyclic group is isomorphic to the integers or to a finite group of residues modulo an integer.
Group (mathematics) : A set with an associative binary operation, an identity element, and an inverse for every element. Cyclicity is a property of groups, defined through repeated application of their operation.
Integer powers : Repeated products of an element, including the identity power and powers of its inverse. A cyclic group consists exactly of the integer powers of a chosen generator.
Direct product of groups : A group built from tuples of elements of several groups, with operations applied componentwise. A product of cyclic groups need not be cyclic, unlike a single cyclic factor.
Modular arithmetic : Arithmetic on residue classes, where integers differing by a fixed modulus are treated as equivalent. Residue classes modulo n form the additive cyclic group of order n.
Group generator : An element whose powers generate a group, or a subset whose elements generate it together. A cyclic group has a generating set consisting of just one element.
Cyclic group of integers : The additive group of integers, generated by 1 or by −1. It is the basic infinite cyclic group and models every infinite cyclic group.
Dihedral group : The group of symmetries of a regular polygon, generated by rotations and reflections. For polygons with at least three sides, reflections make the full symmetry group noncyclic.
Roots of unity : Complex numbers whose positive integer powers equal 1. The nth roots of unity form a cyclic group under multiplication.
Order of an element : The least positive integer whose power of an element is the identity, when such an integer exists. A generator’s order determines whether its cyclic group is finite and how many elements it contains.
Finite cyclic group : A cyclic group with finitely many elements, isomorphic to the integers modulo its order. A generator’s powers repeat after its finite order, producing this standard form.
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