Rearrangement Theorem

Each row and each column in the Group multiplication table lists each of the Group elements once and only once. From this, it follows that no two elements may be in the identical location in two rows or two columns. Thus, each row and each column is a rearranged list of the Group elements. Stated otherwise, given a Group of $n$ distinct elements $(I, a, b, c, \ldots, n)$, the set of products $(aI, a^2, ab, ac, \ldots, an)$ reproduces the $n$ original distinct elements in a new order.

See also Group