Lets start with by defining what an order of an element is.

Defn. Let 𝐆 be a group. The order of the element 𝐠 ∈ 𝐆 means the Smallest positive integer (ℤ+) 𝒏, such that 𝐠𝒏 = 1. (It’s possible that there is no such 𝒏 and in that case the order of 𝐠 is ). We write the order of 𝐠 as: π—ˆ(𝐠).

Thm. Suppose 𝐠𝒏 = 1 in 𝐆, then π—ˆ(𝐠)βŽœπ’

Proof. Let 𝐝 = π—ˆ(𝐠).

𝒏 = 𝐝πͺ + 𝐫 where 0 ≀ 𝐫 < 𝐝
𝐠𝐫 = 𝐠𝒏 - 𝐝πͺ = (𝐠𝒏) (𝐠𝐝)- πͺ
(𝐠𝒏) = 1
(𝐠𝐝) = 1
𝐠𝐫 = 1

So, 𝐫 must be 0, since 𝐫 < 𝐝 and 𝐝 was the smallest.

More coming soon…