GAP (HAP) - Chapter 23: G-Outer Groups

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23 G-Outer Groups

### 23 G-Outer Groups

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

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