Some of the group axioms are easy to check. The entries within the table are those outside, so the set is closed under the operation. The first row and column inside the table copy the row and column at the top and side, so 1 is a two-sided identity. A 1 appears in each row and each column of the table, and overall the 1 s appear symmetrically in the table, so each element has a two-sided inverse. The table is a latin square, so that, for given a and b, the equations ax = b and ya = b have unique solutions for x and y.