Group Element to Power of Group Order is Identity
Theorem
Given \(g \in G\) with \(|G| = n\) for finite group \(G\)
\[ g^{|G|} = \mathrm{id}_G\]
Specific examples of this fact being useful include Fermat's little theorem and more generally Euler's theorem.
Of course, in an additive group, we use the notation
\[ |G|g = \mathrm{id}_G.\]
Proof
This fact follows simply from the fact that the order of each group element divides the order of the group.
That is for any \(g \in G\), since \(G\) is finite, and the powers up to order in group are distinct, \(g\) must have finite order (there are not infinitely many unique elements). Then since \(\mathrm{ord}(G) \mid |G|\), we have that \(\mathrm{ord}(G) \cdot k = |G|\) for some integer \(k\). Then
\[ g^{\mathrm{ord}(G) \cdot k} = (g^{\mathrm{ord}(G)})^k = (\mathrm{id}_G)^k = \mathrm{id}_G.\]