# Group with operators > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Group_with_operators > Markdown URL: https://mediated.wiki/source/Group_with_operators.md > Source: https://en.wikipedia.org/wiki/Group_with_operators > Source revision: 1347375218 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Concept in mathematics regarding sets operating on groups}} In [abstract algebra](/source/abstract_algebra), a branch of [mathematics](/source/mathematics), a '''group with operators''' or Ω-'''group''' is an [algebraic structure](/source/algebraic_structure) that can be viewed as a [group](/source/group_(mathematics)) together with a [set](/source/set_(mathematics)) Ω that operates on the elements of the group in a special way. Groups with operators were extensively studied by [Emmy Noether](/source/Emmy_Noether) and her school in the 1920s. She employed the concept in her original formulation of the three [Noether isomorphism theorems](/source/Noether_isomorphism_theorems). {{Algebraic structures|Module}} == Definition == A '''group with operators''' (G, \Omega) can be defined{{sfn|Bourbaki|1974|p=31}} as a group G = (G, \cdot) together with an action of a set \Omega on G: : \Omega \times G \rightarrow G : (\omega, g) \mapsto g^\omega that is [distributive](/source/distributive_property) relative to the group law: : (g \cdot h)^\omega = g^\omega \cdot h^\omega. For each \omega \in \Omega , the map g \mapsto g^\omega is then an [endomorphism](/source/Group_homomorphism) of ''G''. From this, it results that a Ω-group can also be viewed as a group ''G'' with an [indexed family](/source/indexed_family) \left(u_\omega\right)_{\omega \in \Omega} of endomorphisms of ''G''. \Omega is called the '''operator domain'''. The associate endomorphisms{{sfn|Bourbaki|1974|pp=30–31}} are called the '''homotheties''' of ''G''. Given two groups ''G'', ''H'' with same operator domain \Omega, a '''homomorphism''' of groups with operators from (G, \Omega) to (H, \Omega) is a [group homomorphism](/source/group_homomorphism) \phi: G \to H satisfying : \phi\left(g^\omega\right) = (\phi(g))^\omega for all \omega \in \Omega and g \in G. A [subgroup](/source/subgroup) ''S'' of ''G'' is called a '''stable subgroup''', '''\Omega-subgroup''' or '''\Omega-invariant subgroup''' if it respects the homotheties, that is : s^\omega \in S for all s \in S and \omega \in \Omega. == Category-theoretic remarks == In [category theory](/source/category_theory), a '''group with operators''' can be defined{{sfn|Mac Lane|1998|p=41}} as an [object](/source/object_(category_theory)) of a [functor category](/source/functor_category) '''Grp'''''M'' where ''M'' is a [monoid](/source/monoid) (i.e. a [category](/source/category_(mathematics)) with one object) and '''Grp''' denotes the [category of groups](/source/category_of_groups). This definition is equivalent to the previous one, provided \Omega is a monoid (if not, we may expand it to include the [identity](/source/identity_function) and all [compositions](/source/function_composition)). A [morphism](/source/morphism) in this category is a [natural transformation](/source/natural_transformation) between two [functor](/source/functor)s (i.e., two groups with operators sharing same operator domain ''M''{{hairsp}}). Again we recover the definition above of a homomorphism of groups with operators (with ''f'' the [component](/source/natural_transformation) of the natural transformation). A group with operators is also a mapping :\Omega \rightarrow \operatorname{End}_\mathbf{Grp}(G), where \operatorname{End}_\mathbf{Grp}(G) is the set of group endomorphisms of ''G''. == Examples == * Given any group ''G'', (''G'', ∅) is trivially a group with operators * Given a [module](/source/module_(mathematics)) ''M'' over a [ring](/source/ring_(mathematics)) ''R'', ''R'' acts by [scalar multiplication](/source/scalar_multiplication) on the underlying [abelian group](/source/abelian_group) of ''M'', so (''M'', ''R'') is a group with operators. * As a special case of the above, every [vector space](/source/vector_space) over a [field](/source/field_(mathematics)) ''K'' is a group with operators (''V'', ''K''). ==Applications== The [Jordan–Hölder theorem](/source/Jordan%E2%80%93H%C3%B6lder_theorem) also holds in the context of groups with operators. The requirement that a group have a [composition series](/source/composition_series) is analogous to that of [compactness](/source/compact_space) in [topology](/source/topology), and can sometimes be too strong a requirement. It is natural to talk about "compactness relative to a set", i.e. talk about composition series where each ([normal](/source/Normal_subgroup)) subgroup is an operator-subgroup relative to the operator set ''X'', of the group in question. ==See also== * [Group action](/source/Group_action_(mathematics)) ==Notes== {{reflist}} == References == *{{cite book | last=Bourbaki | first=Nicolas |author-link=Nicolas Bourbaki | title=Elements of Mathematics : Algebra I Chapters 1–3 | publisher=Hermann | year=1974 | isbn=2-7056-5675-8 | url-access=registration | url=https://archive.org/details/algebra0000bour }} *{{cite book | last=Bourbaki | first=Nicolas | title=Elements of Mathematics : Algebra I Chapters 1–3 | publisher=Springer-Verlag | year=1998 | isbn=3-540-64243-9}} *{{cite book | last=Mac Lane | first=Saunders |author-link=Saunders Mac Lane | title=[Categories for the Working Mathematician](/source/Categories_for_the_Working_Mathematician) | publisher=Springer-Verlag | year=1998 | isbn=0-387-98403-8}} Category:Group actions Category:Universal algebra --- Adapted from the Wikipedia article [Group with operators](https://en.wikipedia.org/wiki/Group_with_operators) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Group_with_operators?action=history)). 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