G_{2} F_{4} E_{6} E_{7} E_{8}
Lorentz group
Poincaré group
Conformal group
Diffeomorphism group
Loop group
Infinitedimensional 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).
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Definition
A group G is called cyclic if there exists an element g in G such that G = <g> = { g^{n}  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 = { g^{0}, g^{1}, g^{2}, g^{3}, g^{4}, g^{5} } is a group, then g^{6} = g^{0}, 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 g^{1}·g^{2} = g^{3}, and 2 + 5 = 1 (mod 6) corresponds to g^{2}·g^{5} = g^{7} = g^{1}, and so on. One can use the isomorphism φ defined by φ(g^{i}) = 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 g^{n} 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.
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