Cyclic Groups Lecture Notes
Cyclic Groups M. Velrajan Consider the group G = {1, 𝜔, 𝜔 2 , … , 𝜔 n-1 } of all n th roots of unity under multiplication, where 𝜔 = e 2𝝅i/n , 𝜔 n = 1. Here 𝜔 n+1 = 𝜔, 𝜔 n+2 = 𝜔 2 , and 𝜔 n+k = 𝜔 k for any k ≥ 1. Also 𝜔 -1 = 𝜔 n-1 , 𝜔 -2 = 𝜔 n-2 , 𝜔 -n = 𝜔 n-n = 1, 𝜔 -(n+1) = 𝜔 n-1 , …. We note that all integer powers of 𝜔 are in G. Such groups are called cyclic groups. Before defining a cyclic group, we prove the following. Theorem Let G be a group and a ∈ G. Then H = {a n / n ∈ Z} is a subgroup of G. Proof Clearly H is a nonempty subset of G. For x = a m and y = a n ∈ H, xy -1 = a m (a n ) -1 = a m a -n = a m-n ∈ H. Hence H is a subgroup of G. Definition Let G be a group and a ∈ G. Then the subgroup {a n / n ∈ Z} is called the cyclic subgroup of G generated by...