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).
|||A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)|
The usual notion of a $p$-group (i.e., a group of order a power of $p$) is not to be mixed up with that of an $n$-group in the above sense.
|[a1]||D. Balci, "Zur Theorie der topologischen $n$-Gruppen" , Minerva , Munich (1981)|
|[a2]||S.A. Rusakov, "The subgroup structure of Dedekind $n$-ary groups" , Finite groups (Proc. Gomel. Sem.) , Minsk (1978) pp. 81–104 (In Russian)|
|[a3]||S.A. Rusakov, "On the theory of nilpotent $n$-ary groups" , Finite groups (Proc. Gomel. Sem.) , Minsk (1978) pp. 104–130 (In Russian)|
N-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=N-group&oldid=34737