Definition:Additive Group

Definition

Additive Group of Integers

The additive group of integers $\struct {\Z, +}$ is the set of integers under the operation of addition.

Additive Group of Integer Multiples

Let $n \in \Z_{>0}$.

The additive group $\left({n \Z, +}\right)$ of integer multiples of $n$ is the set of integer multiples of $n$ under the operation of addition.

Additive Group of Integers Modulo $m$

Let $m \in \Z$ such that $m > 1$.

The additive group of integers modulo $m$, denoted $\struct {\Z_m, +_m}$, is the set of integers modulo $m$ under the operation of addition modulo $m$.

Additive Group of Rational Numbers

The additive group of rational numbers $\struct {\Q, +}$ is the set of rational numbers under the operation of addition.

Additive Group of Real Numbers

The additive group of real numbers $\struct {\R, +}$ is the set of real numbers under the operation of addition.

Additive Group of Complex Numbers

The additive group of complex numbers $\struct {\C, +}$ is the set of complex numbers under the operation of addition.

Abstract Algebra

Additive Group of Ring

The group $\struct {R, +}$ is known as the additive group of $R$.

Additive Group of Field

The group $\struct {F, +}$ is known as the additive group of $F$.

Also defined as

Some sources make special issue of the nature of a group when its underlying set is a subset of, or derived directly from, numbers themselves.

In such treatments, a group whose operation is addition is then referred to as an additive group.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ we consider all groups, whatever their nature, to be instances of the same abstract concept, and therefore make no such distinction.

Some sources confuse and muddy the water still further by calling an additive group any group whose notation is such that it uses $+$ as the symbol to denote the group operation and use $0$ to denote the identity.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): add: 2.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): additive group