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Cyclic groups

A cyclic group is a group all of whose elements are powers of a particular element a {\displaystyle a}. In multiplicative notation, the elements of the group are …, a − 3, a − 2, a − 1, a 0, a, a 2, a 3, …, {\displaystyle \dots,a^{-3},a^{-2},a^{-1},a^{0},a,a^{2},a^{3},\dots,} where a 2 {\displaystyle a^{2}} means a ⋅ a {\displaystyle a\cdot a}, a − 3 {\displaystyle a^{-3}} stands for a − 1 ⋅ a − 1 ⋅ a − 1 = (a ⋅ a ⋅ a) − 1 {\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1}}, etc. Such an element a {\displaystyle a} is called a generator or a primitive element of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as …, (− a) + (− a), − a, 0, a, a + a, …. {\displaystyle \dots,(-a)+(-a),-a,0,a,a+a,\dots.}

In the groups (Z / n Z, +) {\displaystyle (\mathbb {Z} /n\mathbb {Z},+)} introduced above, the element 1 {\displaystyle 1} is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1 {\displaystyle 1}. Any cyclic group with n {\displaystyle n} elements is isomorphic to this group. A second example for cyclic groups is the group of n {\displaystyle n} th complex roots of unity, given by complex numbers z {\displaystyle z} satisfying z n = 1 {\displaystyle z^{n}=1}. These numbers can be visualized as the vertices on a regular n {\displaystyle n} -gon, as shown in blue in the image for n = 6 {\displaystyle n=6}. The group operation is multiplication of complex numbers. In the picture, multiplying with z {\displaystyle z} corresponds to a counter-clockwise rotation by 60°. From field theory, the group F p × {\displaystyle \mathbb {F} _{p}^{\times }} is cyclic for prime p {\displaystyle p}: for example, if p = 5 {\displaystyle p=5}, 3 {\displaystyle 3} is a generator since 3 1 = 3 {\displaystyle 3^{1}=3}, 3 2 = 9 ≡ 4 {\displaystyle 3^{2}=9\equiv 4}, 3 3 ≡ 2 {\displaystyle 3^{3}\equiv 2}, and 3 4 ≡ 1 {\displaystyle 3^{4}\equiv 1}.

Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element a {\displaystyle a}, all the powers of a {\displaystyle a} are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to (Z, +) {\displaystyle (\mathbb {Z},+)}, the group of integers under addition introduced above. As these two prototypes are both abelian, so are all cyclic groups.

The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.