Question

If g is a finite group, then show that there exists a positive integer n such that an = e for all a g.

Answer 1

Since G is finite, then G has finite number of elements. Assume G={a1,a2,a3,⋯,aN} for some positive integer N. For some ai∈G, consider the sequence a1i,a2i,a3i,⋯

. All these elements are in G (closed under binary operation). since G is finite, then there must be repetitions in the above sequence, i.e. aji=aki for some positive integers j and k with j>k (without any loss of generality). i.e aj−ki=e i.e anii=e.