Additive group in the context of Ring (mathematics)

Addition, usually denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as "3 + 2 = 5", which is read as "three plus two equals five".

Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, and elements of additive groups.

In abstract algebra, a cyclic group or monogenous group is a group, denoted C_n (also frequently ${\displaystyle \mathbb {Z} }$_n or Z_n, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group.

Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.

In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does H. Furthermore, H itself is both a left coset and a right coset. The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by [G : H].

Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group G, the number of elements of every subgroup H of G divides the number of elements of G. Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes.