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Definition and notation

For any element g in any group G, one can form the subgroup that consists of all its integer powers: ⟨ g ⟩ = { g | k ∈ Z }, called the cyclic subgroup generated by g. The order of g is |⟨ g ⟩|, the number of elements in ⟨ g ⟩, conventionally abbreviated as | g |, as ord(g), or as o(g). That is, the order of an element is equal to the order of the cyclic subgroup that it generates.

A cyclic group is a group which is equal to one of its cyclic subgroups: G = ⟨ g ⟩ for some element g, called a generator of G.

For a finite cyclic group G of order n we have G = { e, g, g,..., g }, where e is the identity element and g = g whenever i ≡ j (mod n); in particular g = g = e, and g = g. An abstract group defined by this multiplication is often denoted C, and we say that G is isomorphic to the standard cyclic group C. Such a group is also isomorphic to Z / n Z, the group of integers modulo n with the addition operation, which is the standard cyclic group in additive notation. Under the isomorphism χ defined by χ (g) = i the identity element e corresponds to 0, products correspond to sums, and powers correspond to multiples.

For example, the set of complex 6th roots of unity: G = { ± 1, ± (1 2 + 3 2 i), ± (1 2 − 3 2 i) } {\displaystyle G=\left\{\pm 1,\pm {\left({\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i\right)},\pm {\left({\tfrac {1}{2}}-{\tfrac {\sqrt {3}}{2}}i\right)}\right\}} forms a group under multiplication. It is cyclic, since it is generated by the primitive root z = 1 2 + 3 2 i = e 2 π i / 6: {\displaystyle z={\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i=e^{2\pi i/6}:} that is, G = ⟨ z ⟩ = { 1, z, z, z, z, z } with z = 1. Under a change of letters, this is isomorphic to (structurally the same as) the standard cyclic group of order 6, defined as C = ⟨ g ⟩ = { e, g, g, g, g, g } with multiplication g · g = g, so that g = g = e. These groups are also isomorphic to Z /6 Z = {0, 1, 2, 3, 4, 5} with the operation of addition modulo 6, with z and g corresponding to k. For example, 1 + 2 ≡ 3 (mod 6) corresponds to z · z = z, and 2 + 5 ≡ 1 (mod 6) corresponds to z · z = z = z, and so on. Any element generates its own cyclic subgroup, such as ⟨ z ⟩ = { e, z, z } of order 3, isomorphic to C and Z /3 Z; and ⟨ z ⟩ = { e, z, z = z, z = z, z = z, z = z } = G, so that z has order 6 and is an alternative generator of G.

Instead of the quotient notations Z / n Z, Z /(n), or Z / n, some authors denote a finite cyclic group as Z, but this clashes with the notation of number theory, where Z denotes a p -adic number ring, or localization at a prime ideal.

On the other hand, in an infinite cyclic group G = ⟨ g ⟩, the powers g give distinct elements for all integers k, so that G = {..., g, g, e, g, g,... }, and G is isomorphic to the standard group C = C and to Z, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, and the name "cyclic" may be misleading.

To avoid this confusion, Bourbaki introduced the term monogenous group for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group".