# Order of an Element

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

** Defn.** Let π be a group. The order of the element π β π means the

*Smallest positive*integer (ℤ+) π, such that π

^{π}= 1. (Itβs possible that there is no such π and in that case the order of π is ∞). We write the order of π as: π(π ).

** Thm.** Suppose π

^{π}= 1 in π, then π(π )βπ

** Proof.** Let π = π(π ).

^{π«}= π

^{π - ππͺ}= (π

^{π}) (π

^{π})

^{- πͺ}

^{π}) = 1

^{π}) = 1

^{π«}= 1

So, π« must be 0, since π« < π and π was the smallest.

More coming soonβ¦