If we identify the free group on the singleton with the additive group of integers, then the infinite cyclic group is itself, while the finite cyclic group of order is , that is modulo (the normal subgroup generated by) the integer . Of course, itself is also . One could also consider for negative , but this is the same as .
The cyclic group of order may also be identified with a subgroup of the multiplicative group of complex numbers (or algebraic numbers): the group of th roots of . For , we may pick any non-zero complex number (or even something else) that is not a root of (but there is no standard choice) and take the subgroup generated by it.
Besides ‘’, the cyclic group of order may also be denoted in other ways: some more complicated variation of ‘’ (to put ‘the [normal] subgroup generated by’ explicitly in the notation), or else the simplified form ‘’ (which however conflicts with notation for the -adic integers). When written multiplicatively, it may be denoted ‘’ (note the font change) or ‘’; either letter here stands for ‘cyclic’ in one language or another. (It is a coincidence that the German words ‘Zahl’, which gives us ‘’, and ‘zyklisch’, which gives us ‘’, begin with the same letter.)
Besides ‘’, the infinite cyclic group may also be denoted in other ways: some variation of ‘’ to indicate that we are using addition of integers, or any of the above notations with either ‘’ or ‘’ in place of ‘’ (depending on whether we think of it as modulo or the cyclic group with order ).
When written additively, the notation for the elements of a cyclic group are usually just the notation for integers; for the finite cyclic group of order , we use the natural number less than . In the finite case, we may also use brackets or some other notation to indicate equivalence classes. When written multiplicatively, any letter (‘’, ‘’, ‘’, ‘’, etc) may be taken to stand for the generating element; then any other element is a power of this generator. When thought of as a multiplicative group of complex numbers, one generator is , and the notation may reflect that.
Let be a cyclic group, and let be a generator of . Then there is a unique ring structure on (making the original group the additive group of the ring) such that is the multiplicative identity .
If we identify with the additive group and pick (the equivalence class of) the integer for , then the resulting ring is precisely the quotient ring .
In this way, a cyclic group equipped with the extra structure of a generator is the same thing (in the sense that their groupoids are equivalent) as a ring with the extra property that the underlying additive group is cyclic.
For , the number of ring structures on the cyclic group , which is the same as the number of generators, is , the Euler totient? of ; the generators are those that are relatively prime to . While , otherwise (another way to see that we have a structure and not just a property). For itself, there are two ring structures, since and are the generators (and these are relatively prime to ).
Moreover, the lattice of subgroups of is equivalently the dual of the lattice of natural numbers ordered by divisibility.
(e.g Aluffi 09, pages 83-84)
This is a special case of the fundamental theorem of finitely generated abelian groups. See there for more.
Relevant for Dijkgraaf-Witten theory is the fact
See at finite abelian group for details.
Paolo Aluffi, Algebra: Chapter 0, Part 0, 2009
Cyclic groups pdf
Joseph A. Gallian, Fundamental Theorem of Cyclic Groups, Contemporary Abstract Algebra, p. 77, (2010)