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Concept
Cyclic group
Summary
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted Cn, that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group.
Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.
Every cyclic group of prime order is a simple group, which cannot be broken down into smaller groups. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.
For any element g in any group G, one can form the subgroup that consists of all its integer powers: ⟨g⟩ = , 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 = , where e is the identity element and gi = gj whenever i ≡ j (mod n); in particular gn = g0 = e, and g−1 = gn−1. An abstract group defined by this multiplication is often denoted Cn, and we say that G is isomorphic to the standard cyclic group Cn.
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