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

Let be a group with multiplication operation ; let be an arbitrary integer. Then an -ary operation on the set can be defined as follows:

The resulting -group is called the -group determined by the group . Necessary and sufficient conditions for an -group to be of this form are known [1]. Any -group is imbeddable in such an -group (Post's theorem).


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


The usual notion of a -group (i.e., a group of order a power of ) is not to be mixed up with that of an -group in the above sense.


[a1] D. Balci, "Zur Theorie der topologischen -Gruppen" , Minerva , Munich (1981)
[a2] S.A. Rusakov, "The subgroup structure of Dedekind -ary groups" , Finite groups (Proc. Gomel. Sem.) , Minsk (1978) pp. 81–104 (In Russian)
[a3] S.A. Rusakov, "On the theory of nilpotent -ary groups" , Finite groups (Proc. Gomel. Sem.) , Minsk (1978) pp. 104–130 (In Russian)
How to Cite This Entry:
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This article was adapted from an original article by V.D. Belousov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article