If G is a finite group, prove that, given a is an element of G, there is a positive integer n, depending on a, such that aⁿ = e.
a) By Lagrange's theorem, n divides the order of G.
b) By the definition of a group, aⁿ = e for some n in G.
c) By the pigeonhole principle, there are infinitely many n such that aⁿ = e.
d) By the definition of a finite group, G has a finite number of elements, so there exists n such that aⁿ = e.