  
  [1X22 [33X[0;0YG-Outer Groups[133X[101X
  
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XGOuterGroup(E,N)[110X[133X
  [33X[0;0Y[10XGOuterGroup()[110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XGOuterGroupHomomorphismNC(A,B,phi)[110X[133X
  [33X[0;0Y[10XGOuterGroupHomomorphismNC()[110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XGOuterHomomorphismTester(A,B,phi)[110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XCentre(A)[110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XDirectProductGog(A,B)[110X[133X
  [33X[0;0Y[10XDirectProductGog(Lst)[110X[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
  
