can be constructed in a similar fashion to a multiplication table. If the group is under the the entry of this table contains the product . The group is abelian if and only if this table is symmetric about the main diagonal. This is true since the group is abelian iff for all , which is iff the entry of the table equals the entry for all , i.e. the table is symmetric about the main diagonal. Examples For the integers and the operation addition , denoted , the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer has an additive inverse, , and the addition operation is commutative since for any two integers and . Every cyclic group is abelian, because if , are in , then . Thus the integers, , form an abelian group under addition, as do the integers modulo , . Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication. Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order. The concepts of abelian group and -module agree. More specifically, every -module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers in a unique way. In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of rotation matrices. Historical remarks Camille Jordan named abelian groups after Norwegian mathematician Niels Henrik Abel, because Abel found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals. Properties If is a natural number and is an element of an abelian group written additively, then can be defined as ( summands) and . In this way, becomes a module over the ring of integers. In