Group order

The order of an element and its inverse are the same

Given a group 𝐺 and an element 𝑎 ∈𝐺 with order |𝑎| =𝑛, the order of the inverse ∣𝑎−1∣ =𝑛. group

Proof

By the uniqueness of the inverse, 𝑎𝑛 =𝑒 iff 𝑎−𝑛 =𝑒. Therefore the orders of 𝑎 and 𝑎−1 must be equal, since neither can have a lower order than the other.

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