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Difference between revisions of "N-group"

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A generalization of the concept of a [[Group|group]] to the case of an -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