Prove that a finite group is abelian if and only if its group table is a symmetric matrix.

If is a finite group, there exists a bijective indexing map for some natural number ; we can then define the group table of induced by to be a matrix with .

Suppose is a finite abelian group. Then for all we have . Hence is symmetric.

Suppose is a symmetric matrix. Then for all we have , so that is abelian.