  
  [1X23 [33X[0;0YG-Outer Groups[133X[101X
  
  
  [1X23.1 [33X[0;0Y [133X[101X
  
  [1X23.1-1 GOuterGroup[101X
  
  [33X[1;0Y[29X[2XGOuterGroup[102X( [3XE[103X, [3XN[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGOuterGroup[102X(  ) [32X function[133X
  
  [33X[0;0YInputs  a  group  [22XE[122X  and  normal subgroup [22XN[122X. It returns [22XN[122X as a [22XG[122X-outer group
  where [22XG=E/N[122X.[133X
  
  [33X[0;0YThe  function  can  be used without an argument. In this case an empty outer
  group  [22XC[122X  is returned. The components must be set using SetActingGroup(C,G),
  SetActedGroup(C,N) and SetOuterAction(C,alpha).[133X
  
  [33X[0;0Y[12XExamples:[112X   1   ([7X../tutorial/chap6.html[107X) ,  2  ([7X../tutorial/chap7.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutCoefficientSequence.html[107X) ,                   4
  ([7X../www/SideLinks/About/aboutGouter.html[107X) [133X
  
  [1X23.1-2 GOuterGroupHomomorphismNC[101X
  
  [33X[1;0Y[29X[2XGOuterGroupHomomorphismNC[102X [32X global variable[133X
  [33X[1;0Y[29X[2XGOuterGroupHomomorphismNC[102X [32X global variable[133X
  
  [33X[0;0YInputs  G-outer  groups  [22XA[122X  and  [22XB[122X  with  common  acting  group, and a group
  homomorphism   phi:ActedGroup(A)   -->   ActedGroup(B).   It   returns   the
  corresponding  G-outer  homomorphism  PHI:A--> B. No check is made to verify
  that phi is actually a group homomorphism which preserves the G-action.[133X
  
  [33X[0;0YThe  function  can  be used without an argument. In this case an empty outer
  group homomorphism [22XPHI[122X is returned. The components must then be set.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X23.1-3 GOuterHomomorphismTester[101X
  
  [33X[1;0Y[29X[2XGOuterHomomorphismTester[102X( [3XA[103X, [3XB[103X, [3Xphi[103X ) [32X function[133X
  
  [33X[0;0YInputs  G-outer  groups  [22XA[122X  and  [22XB[122X  with  common  acting  group, and a group
  homomorphism  phi:ActedGroup(A) --> ActedGroup(B). It tests whether phi is a
  group homomorphism which preserves the G-action.[133X
  
  [33X[0;0YThe  function  can  be used without an argument. In this case an empty outer
  group homomorphism [22XPHI[122X is returned. The components must then be set.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X23.1-4 Centre[101X
  
  [33X[1;0Y[29X[2XCentre[102X( [3XA[103X ) [32X function[133X
  
  [33X[0;0YInputs   G-outer   group  [22XA[122X  and  returns  the  group  theoretic  centre  of
  ActedGroup(A) as a G-outer group.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap6.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutParallel.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutSchurMultiplier.html[107X) ,                       4
  ([7X../www/SideLinks/About/aboutGouter.html[107X) ,                                5
  ([7X../www/SideLinks/About/aboutLieCovers.html[107X) [133X
  
  [1X23.1-5 DirectProductGog[101X
  
  [33X[1;0Y[29X[2XDirectProductGog[102X( [3XA[103X, [3XB[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDirectProductGog[102X( [3XLst[103X ) [32X function[133X
  
  [33X[0;0YInputs  G-outer  groups  [22XA[122X and [22XB[122X with common acting group, and returns their
  group-theoretic  direct  product as a G-outer group. The outer action on the
  direct product is the diagonal one.[133X
  
  [33X[0;0YThe function also applies to a list Lst of G-outer groups with common acting
  group.[133X
  
  [33X[0;0YFor  a  direct product D constructed using this function, the embeddings and
  projections  can  be  obtained  (as  G-outer  group homomorphisms) using the
  functions Embedding(D,i) and Projection(D,i).[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
