An Ω\Omega-group is a group equipped with additional algebraic operations (of signature Ω\Omega) that distribute over the group operations.


We take an algebraic structure in a traditional sense as a set Ω\Omega with a (not necessarily finite) number of operations α\alpha of various arity and satisfying some axioms (not necessarily of first order). That is, we are discussing objects of an equationally presentable or algebraic category.

An Ω\Omega-group is an algebraic structure which amounts to a group (usually written additively but not necessarily commutative) together with a set Ω\Omega of operations of any arity, such that for each nn-ary operation αΩ\alpha \in \Omega, distributivity holds in each variable over the group operations:

α(x 1,,x j+y j,,x n)=α(x 1,,x j,,x n)+α(x 1,,y j,,x n). \alpha(x_1,\ldots,x_j + y_j,\ldots,x_n)= \alpha(x_1,\ldots,x_j,\ldots,x_n) + \alpha(x_1,\ldots,y_j,\ldots,x_n) .

(This states only distributivity over addition; however, distributivity over all other group operations follows.)


The classical examples are of course groups (where Ω\Omega is empty), rngs (where Ω\Omega consists of only multiplication), and rings (where Ω\Omega consists of multiplication and the nullary operator that gives the multiplicative identity). Given a fixed ground ring kk, the modules over kk form another example: each element of kk gives a unary multiplication operation.

The older term group with operators is traditionally used for Ω\Omega-groups when only unary operations are considered (as in the case of modules).


The general theory of Ω\Omega-groups is similar to the basics of group and ring theory, including normal subgroups / ideals / submodules, quotient? Ω\Omega-groups, Noether’s isomorphism theorem?s, etc. For example, the Jordan–Holder theorem holds: if there is a composition series, then every two composition series are equivalent up to permutation of factors. An obvious horizontal categorification of Ω\Omega-groups is also interesting.

Note that Ω\Omega is a capital Greek letter; ω\omega-group is rather a synonym for (for some people strict) \infty-groupoid with a single object, hence nothing to do with Ω\Omega-groups.

Ω\Omega-groups form a protomodular category.


  • Wikipedia, Group with operators.

  • N. Bourbaki, Algebra I, ch. 1-3.

  • E. I. Khukhro, Local nilpotency in varieties of groups with operators, Russ. Acad. Sci. Sbornik Mat. 78 379, 1994. (doi)

  • Grace Orzech, Obstruction theory in algebraic categories I, II, J. Pure Appl. Algebra 2 (1972) 287-340, 315–340.

Revised on November 3, 2016 05:59:41 by David Corfield (