# Definition:Additive Group

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## 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 $m$ $\struct {\Z_m, +_m}$ 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 $\left({\Q, +}\right)$ 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.