# nLab n-ary group

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

A generalization of associative quasigroup to n-ary operations.

## Definition

Given a natural number $n$, an $n$-ary group is a set $S$ that both an n-ary semigroup and an n-ary quasigroup.

###### Remark

A binary group is not a group. Binary groups are associative quasigroups, as they can be empty.

## Properties

###### Definition

An n-ary identity element is an element $e$ such that any string of $n$ elements consisting of all $e$‘s, apart from one place, is mapped to the element at that place.

###### Theorem

Every n-ary group with an n-ary identity element is a group, with the n-ary group operation being simply repeated application of a group’s binary operation.

###### Remark

The above theorem is why most authors typically do not require the existence of $n$-ary identity elements in the definition of $n$-ary groups, and why binary groups are just associative quasigroups.