nLab n-ary group




A generalization of associative quasigroup to n-ary operations.


Given a natural number nn, an nn-ary group is a set SS that both an n-ary semigroup and an n-ary quasigroup.


A binary group is not a group. Binary groups are associative quasigroups, as they can be empty.



An n-ary identity element is an element ee such that any string of nn elements consisting of all ee‘s, apart from one place, is mapped to the element at that place.


Every n-ary group with an n-ary identity element is a group, with the n-ary group operation being simply repeated application of a group’s binary operation.


The above theorem is why most authors typically do not require the existence of nn-ary identity elements in the definition of nn-ary groups, and why binary groups are just associative quasigroups.

See also


  • W. Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Mathematische Zeitschrift, vol. 29 (1928), pp. 1-19.

  • E. L. Post, Polyadic groups, Transactions of the American Mathematical Society 48 (1940), 208–350.

  • W. A. Dudek, “On some old and new problems in n-ary groups”, Quasigroups and Related Systems, 8 (2001), 15–36.

  • Wiesław A. Dudek, Remarks to Głazek’s results on n-ary groups, Discussiones Mathematicae. General Algebra and Applications 27 (2007), 199–233.

  • Wiesław A. Dudek and Kazimierz Głazek, Around the Hosszú-Gluskin theorem for n-ary groups, Discrete Mathematics 308 (2008), 486–4876.

  • Wikipedia, n-ary group

Created on August 4, 2022 at 00:48:08. See the history of this page for a list of all contributions to it.