Group with operators

<h2 id="definition">Definition</h2>
A group with operators 
 
 
 
 (
 G
 ,
 Ω
 )
 
 
 {\displaystyle (G,\Omega )}
 
 can be defined<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> as a group 
 
 
 
 G
 =
 (
 G
 ,
 ⋅
 )
 
 
 {\displaystyle G=(G,\cdot )}
 
 together with an action of a set 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 on 
 
 
 
 G
 
 
 {\displaystyle G}
 
:

Ω
        ×
        G
        →
        G
        :
        (
        ω
        ,
        g
        )
        ↦
        
          g
          
            ω
          
        
      
    
    {\displaystyle \Omega \times G\rightarrow G:(\omega ,g)\mapsto g^{\omega }}

that is <a href="/facts/Distributive_property/0LNBrVJl">distributive</a> relative to the group law:

(
        g
        ⋅
        h
        
          )
          
            ω
          
        
        =
        
          g
          
            ω
          
        
        ⋅
        
          h
          
            ω
          
        
        .
      
    
    {\displaystyle (g\cdot h)^{\omega }=g^{\omega }\cdot h^{\omega }.}

For each 
 
 
 
 ω
 ∈
 Ω
 
 
 {\displaystyle \omega \in \Omega }
 
, the application 
 
 
 
 g
 ↦
 
 g
 
 ω
 
 
 
 
 {\displaystyle g\mapsto g^{\omega }}
 
 is then an <a href="/facts/Group_homomorphism/sDguaNYs">endomorphism</a> of G. From this, it results that a Ω-group can also be viewed as a group G with an <a href="/facts/Indexed_family/aFqfzJN0">indexed family</a> 
 
 
 
 
 
 (
 
 u
 
 ω
 
 
 )
 
 
 ω
 ∈
 Ω
 
 
 
 
 {\displaystyle \left(u_{\omega }\right)_{\omega \in \Omega }}
 
 of endomorphisms of G.

 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is called the operator domain. The associate endomorphisms<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> are called the homotheties of G.
Given two groups G, H with same operator domain 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
, a homomorphism of groups with operators from 
 
 
 
 (
 G
 ,
 Ω
 )
 
 
 {\displaystyle (G,\Omega )}
 
 to 
 
 
 
 (
 H
 ,
 Ω
 )
 
 
 {\displaystyle (H,\Omega )}
 
 is a <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a> 
 
 
 
 ϕ
 :
 G
 →
 H
 
 
 {\displaystyle \phi :G\to H}
 
 satisfying

ϕ
        
          (
          
            g
            
              ω
            
          
          )
        
        =
        (
        ϕ
        (
        g
        )
        
          )
          
            ω
          
        
      
    
    {\displaystyle \phi \left(g^{\omega }\right)=(\phi (g))^{\omega }}
  
 for all 
  
    
      
        ω
        ∈
        Ω
      
    
    {\displaystyle \omega \in \Omega }
  
 and 
  
    
      
        g
        ∈
        G
        .
      
    
    {\displaystyle g\in G.}

A <a href="/facts/Subgroup/r91mBgpa">subgroup</a> S of G is called a stable subgroup, 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
-subgroup or 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
-invariant subgroup if it respects the homotheties, that is

s
          
            ω
          
        
        ∈
        S
      
    
    {\displaystyle s^{\omega }\in S}
  
 for all 
  
    
      
        s
        ∈
        S
      
    
    {\displaystyle s\in S}
  
 and 
  
    
      
        ω
        ∈
        Ω
        .
      
    
    {\displaystyle \omega \in \Omega .}

<h2 id="category-theoretic-remarks">Category-theoretic remarks</h2>
In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a group with operators can be defined<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> as an <a href="/facts/Object_(category_theory)/7WzxUThf">object</a> of a <a href="/facts/Functor_category/r7sWdb2h">functor category</a> GrpM where M is a <a href="/facts/Monoid/6AHve7p8">monoid</a> (i.e. a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> with one object) and Grp denotes the <a href="/facts/Category_of_groups/sxSd9Kbc">category of groups</a>. This definition is equivalent to the previous one, provided 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is a monoid (if not, we may expand it to include the <a href="/facts/Identity_function/9AtsP0LC">identity</a> and all <a href="/facts/Function_composition/r5Pvnmth">compositions</a>).
A <a href="/facts/Morphism/Kdpuyz3S">morphism</a> in this category is a <a href="/facts/Natural_transformation/xwPumTHt">natural transformation</a> between two <a href="/facts/Functor/l2u3rIBE">functors</a> (i.e., two groups with operators sharing same operator domain M ). Again we recover the definition above of a homomorphism of groups with operators (with f the <a href="/facts/Natural_transformation/xwPumTHt">component</a> of the natural transformation).
A group with operators is also a mapping

Ω
        →
        
          End
          
            
              G
              r
              p
            
          
        
        ⁡
        (
        G
        )
        ,
      
    
    {\displaystyle \Omega \rightarrow \operatorname {End} _{\mathbf {Grp} }(G),}

where 
 
 
 
 
 End
 
 
 G
 r
 p
 
 
 
 ⁡
 (
 G
 )
 
 
 {\displaystyle \operatorname {End} _{\mathbf {Grp} }(G)}
 
 is the set of group endomorphisms of G.

<h2 id="examples">Examples</h2>
<ul><li>Given any group G, (G, ∅) is trivially a group with operators</li>
<li>Given a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> M over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> R, R acts by <a href="/facts/Scalar_multiplication/jroFDfbM">scalar multiplication</a> on the underlying <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> of M, so (M, R) is a group with operators.</li>
<li>As a special case of the above, every <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K is a group with operators (V, K).</li></ul>
<h2 id="applications">Applications</h2>
The <a href="/facts/Jordan%25E2%2580%2593H%25C3%25B6lder_theorem/3C7CHH3k">Jordan–Hölder theorem</a> also holds in the context of groups with operators. The requirement that a group have a <a href="/facts/Composition_series/3C7CHH3k">composition series</a> is analogous to that of <a href="/facts/Compact_space/d0cgXJH7">compactness</a> in <a href="/facts/Topology/2EqW47cU">topology</a>, 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 (<a href="/facts/Normal_subgroup/aRDaGwLm">normal</a>) subgroup is an operator-subgroup relative to the operator set X, of the group in question.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Group_action_(mathematics)/Vh9VtUUw">Group action</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Bourbaki, Nicolas (1974). <a href="https://archive.org/details/algebra0000bour">Elements of Mathematics : Algebra I Chapters 1–3</a>. Hermann. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 2-7056-5675-8.</li>
<li>Bourbaki, Nicolas (1998). Elements of Mathematics : Algebra I Chapters 1–3. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-64243-9.</li>
<li>Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98403-8.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bourbaki 1974, p. 31. - Bourbaki, Nicolas (1974). Elements of Mathematics : Algebra I Chapters 1–3. Hermann. ISBN 2-7056-5675-8. <a href="https://archive.org/details/algebra0000bour" target="_blank">https://archive.org/details/algebra0000bour</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Bourbaki 1974, pp. 30–31. - Bourbaki, Nicolas (1974). Elements of Mathematics : Algebra I Chapters 1–3. Hermann. ISBN 2-7056-5675-8. <a href="https://archive.org/details/algebra0000bour" target="_blank">https://archive.org/details/algebra0000bour</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Mac Lane 1998, p. 41. - Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer-Verlag. ISBN 0-387-98403-8. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Group with operators open-in-new

Group with operators