cyclic group
A cyclic group is a group you can generate entirely from a single element, just by applying the operation to that element over and over. Picture the hours on a clock face: starting from “1 step” and repeating it, you eventually reach every hour and then loop back to where you began. One starting move was enough to reach the whole structure.
Formally, a group G is cyclic if there is an element g (a generator) such that every element of G is a power g^n for some integer n (or, in additive notation, a multiple n·g). Finite cyclic groups look exactly like the integers mod n under addition; the one infinite cyclic group is the integers under addition, generated by 1.
Every cyclic group is automatically abelian, because powers of the same element always commute (g^m · g^n = g^(m+n) = g^n · g^m). The converse fails: many abelian groups are not cyclic. Cyclic groups are the simplest non-trivial groups and serve as the building blocks for understanding all finite abelian groups.
The integers mod 4 under addition, {0, 1, 2, 3}, are cyclic with generator 1: 1, 1+1 = 2, 1+1+1 = 3, 1+1+1+1 = 0. Repeating one element reaches everything and returns to the identity 0.
ℤ/4ℤ generated by the single element 1.
A generator need not be unique. In the integers mod 5 under addition, both 1 and 2 generate the whole group, while in the integers mod 6 only 1 and 5 do. The elements that generate the group are exactly those whose order equals the size of the group.