Examples
In 2D and 3D the symmetry group for n-fold rotational symmetry is Cn, of abstract group type Zn. In 3D there are also other symmetry groups which are algebraically the same, see Symmetry groups in 3D that are cyclic as abstract group.
Note that the group S1 of all rotations of a circle (the circle group) is not cyclic, since it is not even countable.
The nth roots of unity form a cyclic group of order n under multiplication. e.g., 0 = z3 − 1 = (z − s0)(z − s1)(z − s2) where s = e2πi/3 and a group of {s0, s1, s2} under multiplication is cyclic.
The Galois group of every finite field extension of a finite field is finite and cyclic; conversely, given a finite field F and a finite cyclic group G, there is a finite field extension of F whose Galois group is G.
Read more about this topic: Cyclic Group
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—G.C. (Georg Christoph)
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