Cyclic group

related topics
{math, number, function}
{group, member, jewish}
{math, energy, light}

G2 F4 E6 E7 E8
Lorentz group
Poincaré group
Conformal group
Diffeomorphism group
Loop group
Infinite-dimensional Lie groups O(∞) SU(∞) Sp(∞)

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of g (a multiple of g when the notation is additive).



A group G is called cyclic if there exists an element g in G such that G = <g> = { gn | n is an integer }. Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group G that contains g is G itself suffices to show that G is cyclic.

For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G is cyclic. In fact, G is essentially the same as (that is, isomorphic to) the set { 0, 1, 2, 3, 4, 5 } with addition modulo 6. For example, 1 + 2 = 3 (mod 6) corresponds to g1·g2 = g3, and 2 + 5 = 1 (mod 6) corresponds to g2·g5 = g7 = g1, and so on. One can use the isomorphism φ defined by φ(gi) = i.

For every positive integer n there is exactly one cyclic group (up to isomorphism) whose order is n, and there is exactly one infinite cyclic group (the integers under addition). Hence, the cyclic groups are the simplest groups and they are completely classified.

The name "cyclic" may be misleading: it is possible to generate infinitely many elements and not form any literal cycles; that is, every gn is distinct. (It can be said that it has one infinitely long cycle.) A group generated in this way is called an infinite cyclic group, and is isomorphic to the additive group of integers Z.

Full article ▸

related documents
Direct product
NP (complexity)
Empty set
Greatest common divisor
Polish notation
Ordered pair
Net (mathematics)
Partially ordered set
Binomial theorem
Free group
Affine transformation
Normal space
Graph theory
Grover's algorithm
Knapsack problem
Topological vector space
A* search algorithm
Banach space
Universal quantification
Fundamental theorem of arithmetic
Sheffer stroke
Selection sort
Optimization (mathematics)
Delaunay triangulation
Pell's equation
Document Type Definition
Associative array