A Study on Cyclic Groups |

A cyclic group is a group that isgenerated by a single element, in the sense that every element of the group canbe written as a power of some particular element g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a "generator"of the group. Any infinite cyclic group is isomorphic to Z, the integers with addition as thegroup operation. Any finite cyclic group of order n is isomorphic to Z/nZ, the integers modulo nwith addition as the group operation. Agroup G is called cyclic ifthere exists an element g in G such that G = ⟨g⟩ = { gn | n is an integer }. Since any groupgenerated by an element in a group is a subgroup of that group, showing thatthe only subgroup of a group G that contains g is G itselfsuffices to show that G is cyclic. Forexample, 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 usethe isomorphism χ defined by χ(gi)= i. Forevery positive integer n thereis exactly one cyclic group whose order is n, and there is exactly one infinite cyclic group. Hence, thecyclic groups are the simplest groups and they are completely classified. Thename "cyclic" may be misleading: it is possible to generateinfinitely many elements and not form any literal cycles; that is, every gn is distinct. It can besaid that it has one infinitely long cycle. A group generated in this way iscalled 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. Sincethe cyclic groups are abelian, they are often written additively and denoted Zn. However, this notation can be problematic for numbertheorists because it conflicts with the usual notation for p-adic number rings or localizationat a prime ideal. The quotient notations Z/nZ, Z/n, and Z/(n) are standard alternatives. One may write the groupmultiplicatively, and denote it by Cn,where n is the order (which canbe ∞). For example, g2g4 = g1 in C5,whereas 2 + 4 = 1 in Z/5Z.