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A generalization of the concept of a [[Group|group]] to the case of an $n$-ary operation. An $n$-group is a [[Universal algebra|universal algebra]] with one $n$-ary associative operation that is uniquely invertible at each place (cf. [[Algebraic operation|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.
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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:
 
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:
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Balci,  "Zur Theorie der topologischen $n$-Gruppen" , Minerva , Munich  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.A. Rusakov,  "The subgroup structure of Dedekind $n$-ary groups" , ''Finite groups (Proc. Gomel. Sem.)'' , Minsk  (1978)  pp. 81–104  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.A. Rusakov,  "On the theory of nilpotent $n$-ary groups" , ''Finite groups (Proc. Gomel. Sem.)'' , Minsk  (1978)  pp. 104–130  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Balci,  "Zur Theorie der topologischen $n$-Gruppen" , Minerva , Munich  (1981)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  S.A. Rusakov,  "The subgroup structure of Dedekind $n$-ary groups" , ''Finite groups (Proc. Gomel. Sem.)'' , Minsk  (1978)  pp. 81–104  (In Russian)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  S.A. Rusakov,  "On the theory of nilpotent $n$-ary groups" , ''Finite groups (Proc. Gomel. Sem.)'' , Minsk  (1978)  pp. 104–130  (In Russian)</TD></TR>
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</table>

Latest revision as of 21:09, 21 November 2014


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 [1]. Any $n$-group is imbeddable in such an $n$-group (Post's theorem).

References

[1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)


Comments

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.

References

[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)
How to Cite This Entry:
N-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=N-group&oldid=31010
This article was adapted from an original article by V.D. Belousov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article