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: 𝗈(𝐠).
Thm. Suppose 𝐠𝒏 = 1 in 𝐆, then 𝗈(𝐠)⎜𝒏
Proof. Let 𝐝 = 𝗈(𝐠).
So, 𝐫 must be 0, since 𝐫 < 𝐝 and 𝐝 was the smallest.
More coming soon…