We further drop conditions on the structure of
the binary operations, and study only the
algebraic structure of addition over the
integers.^{1}

A group is an algebra $(G, \circ)$ for which $\circ: G \times G \to G$ is associative, has an identity element in $G$, and has inverse elements. A group is a commutative group (or abelian group) if $\circ$ is commutative.

The number of elements is called the
size (or
order) of the group.^{2}
A group is a finite
group if $G$ is a finite set.

Suppose that $(R, +, \cdot )$ is ring. Then $(R, +)$ is a commutative group. Conversely, suppose $(G, +)$ is a commutative group. Define multiplication on $S$ by $a\cdot b = 0$ for all $a, b \in R$. Then $(S, +, \cdot )$ is a ring, called the zero ring of $(G, +)$. For this reason, it is customary to write $+$ for the operation $\circ$ when handling commutative groups.

Along with the group operation, we call the function which maps an element to its inverse element the group operations.