# 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**