G-Outer Groups

GOuterGroup
GOuterGroup(E,N)
GOuterGroup()

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).

GOuterGroupHomomorphismNC
GOuterGroupHomomorphismNC(A,B,phi)
GOuterGroupHomomorphismNC()

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.

GOuterHomomorphismTester
GOuterHomomorphismTester(A,B,phi)

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.
Centre
Centre(A)

Inputs G-outer group A and returns the group theoretic centre of ActedGroup(A) as a G-outer group.

DirectProductGog
DirectProductGog(A,B)
DirectProductGog(Lst)

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).