Subgroups of Cyclic Groups

In a group G, we denote the (cyclic) group of powers of some g ∈ G by 〈g〉 = {g : k ∈ Z}. If G = 〈g〉, then G itself is cyclic, with g as a generator. Examples of infinite cyclic groups include Z, with (additive) generator 1, and the group 2Z of integral powers of the real number 2, with generator 2. The most basic examples of finite cyclic groups are Z/(m) with (additive) generator 1 and μm = {z ∈ C× : z = 1} with generator e2πi/m = cos(2π/m)+i sin(2π/m). Both have size m. The elements of μm lie at the vertices of a regular m-gon on the unit circle and include 1. This is the fundamental geometric way to picture μm. Don’t confuse the groups Z/(m) and (Z/(m))×. The second, as a set, is a subset of the first, but it is not a subgroup and it is usually not cyclic (see Theorem A.2). For instance, (Z/(15))× is not cyclic. But many of the groups (Z/(m))× are cyclic, and we will take our examples from such groups which are cyclic. Example 1.1. The group (Z/(23))× has size 22, and it is cyclic with 5 as a generator. (The elements 2 and 3 each have order 11, so they are not generators.)