N-group
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).
References
[1] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) |
Comments
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.
References
[a1] | D. Balci, "Zur Theorie der topologischen ![]() |
[a2] | S.A. Rusakov, "The subgroup structure of Dedekind ![]() |
[a3] | S.A. Rusakov, "On the theory of nilpotent ![]() |
N-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=N-group&oldid=20974