Groups

Why

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

Definition

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.

Additive groups

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.

Group Operations

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

1. Future editions will take the genetic approach through polynomial roots, Galois, symmetry and transformations. ↩︎
2. The term order seems more widely used. ↩︎