Cyclic Group - Definition

Definition

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 χ defined by χ(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.

Furthermore, the circle group (whose elements are uncountable) is not a cyclic group—a cyclic group always has countable elements.

Since the cyclic groups are abelian, they are often written additively and denoted Z_n. However, this notation can be problematic for number theorists because it conflicts with the usual notation for p-adic number rings or localization at a prime ideal. The quotient notations Z/nZ, Z/n, and Z/(n) are standard alternatives. We adopt the first of these here to avoid the collision of notation. See also the section Subgroups and notation below.

One may write the group multiplicatively, and denote it by C_n, where n is the order (which can be ∞). For example, g2g4 = g1 in C₅, whereas 2 + 4 = 1 in Z/5Z.