Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 10 / ii / Solution

Since $a^4=I$, multiplication adds exponents modulo four: $a^j{a}^k=a^{j+k\mathrm{mod}4}$. Hence the Cayley table of the cyclic group of order four, with the left factor indexing rows, isThe table is symmetric because the group is Abelian. Its row for each element permutes all four members, as cancellation requires.