  
  [1X17. G-Outer Groups[0X
  
      | [10XGOuterGroup(E,N)[0X [10XGOuterGroup()[0X Inputs a group E and normal subgroup N. It returns N as a G-outer group where G=E/N. The function can be used without an argument. In this case an empty outer group C is returned. The components must be set using SetActingGroup(C,G), SetActedGroup(C,N) and SetOuterAction(C,alpha).                                                                                                                                                                                                 | 
      | [10XGOuterGroupHomomorphismNC(A,B,phi)[0X [10XGOuterGroupHomomorphismNC()[0X Inputs G-outer groups A and B 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. The function can be used without an argument. In this case an empty outer group homomorphism PHI is returned. The components must then be set.                     | 
      | [10XGOuterHomomorphismTester(A,B,phi)[0X Inputs G-outer groups A and B 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. The function can be used without an argument. In this case an empty outer group homomorphism PHI is returned. The components must then be set.                                                                                                                                        | 
      | [10XCentre(A)[0X Inputs G-outer group A and returns the group theoretic centre of ActedGroup(A) as a G-outer group.                                                                                                                                                                                                                                                                                                                                                                                                             | 
      | [10XDirectProductGog(A,B)[0X [10XDirectProductGog(Lst)[0X Inputs G-outer groups A and B 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. The function also applies to a list Lst of G-outer groups with common acting group. For 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). | 
  
