A group axiom. A group axiom. Any operation over a set will have at most one identity element. If the operation is associative then this inverse will be unique. Z and Z n are isomorphic.
Any Abelian group can form a module, a Z -module.
If an operation is both commutative and associative then the orders of the operands are not important. If it is commutative and obeys one of the cancellation laws it will obey the other. If the operation is commutative then the table will be symmetric along its main diagonal, i.e. the diagonal from the upper left corner to the lower right. Every column, and every row, of a group table will be a permutation of the elements of the group. If two sets are congruent, then all their corresponding interior coordinates will be the same.