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A cyclic group is a group generated by a single element a, meaning every element can be written as a power of a. Cyclic groups are always abelian. Every cyclic group of order n is isomorphic to Z/nZ, and every cyclic group of infinite order is isomorphic to Z.
A cyclic group is the simplest kind of group: one generated entirely by repeatedly applying a single element to itself. If you call that element *a*, then the group consists of all "powers" of *a* — that is, a⁰ (the identity), a¹, a², a³, and so on, including negative powers (a⁻¹, a⁻², ...) when they exist. From your work with subgroups, you know that any subset closed under the group operation and inverses forms a subgroup. A cyclic group takes this further: a single element generates the *entire* group, not just a subgroup.
The two archetypes are the integers under addition (the infinite cyclic group, isomorphic to ℤ) and the integers mod n under addition (the finite cyclic group of order n, isomorphic to ℤ/nℤ). In ℤ, the element 1 generates everything: 1, 2 = 1+1, 3 = 1+1+1, and so on, together with −1, −2, ... In ℤ/nℤ, you have a clock with n positions: add 1 repeatedly and you cycle back to 0 after n steps. These are the only cyclic groups up to isomorphism — every cyclic group is one of these two types. Encountering a group and proving it is cyclic therefore pins down its structure completely.
Every cyclic group is abelian (commutative). This follows directly from the definition: since every element is a power of *a*, any two elements aᵐ and aⁿ commute — aᵐ · aⁿ = aᵐ⁺ⁿ = aⁿ · aᵐ. This is a strong structural constraint: if a group is non-abelian, it cannot be cyclic. But the converse fails — an abelian group need not be cyclic. For example, ℤ/2ℤ × ℤ/2ℤ (the Klein four-group) is abelian but not cyclic, because no single element generates all four elements.
One crucial tool is the order of an element: the smallest positive integer *k* such that aᵏ equals the identity. In ℤ/6ℤ, the element 2 has order 3 (since 2+2+2 = 6 ≡ 0), while 1 has order 6. An element generates the entire group ℤ/nℤ if and only if its order equals n, which happens exactly when gcd(element, n) = 1. Counting how many such generators exist leads naturally to Euler's totient function φ(n). Every subgroup of a cyclic group is itself cyclic — a clean structural theorem with no analogue for general groups.