# N-group

2020 Mathematics Subject Classification: Primary: 08A [MSN][ZBL]

A generalization of the concept of a group to the case of an $n$-ary operation. An $n$-group is a universal algebra with one $n$-ary associative operation that is uniquely invertible at each place (cf. Algebraic operation). The theory of $n$-groups for $n\geq 3$ substantially differs from the theory of groups (i.e. $2$-groups). Thus, if $n\geq 3$, an $n$-group has no analogue of the unit element.

Let $\Gamma(\circ)$ be a group with multiplication operation $\circ$; let $n\geq 3$ be an arbitrary integer. Then an $n$-ary operation $\omega$ on the set $\Gamma$ can be defined as follows:

$$a_1\dots a_n\ \omega = a_1\circ\dots\circ a_n$$

The resulting $n$-group is called the $n$-group determined by the group $\Gamma(\circ)$. Necessary and sufficient conditions for an $n$-group to be of this form are known . Any $n$-group is imbeddable in such an $n$-group (Post's theorem).

How to Cite This Entry:
N-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=N-group&oldid=34737
This article was adapted from an original article by V.D. Belousov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article