### 23 G-Outer Groups

#### 23.1

##### 23.1-1 GOuterGroup
 ‣ GOuterGroup( E, N ) ( function )
 ‣ GOuterGroup( ) ( function )

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).

1 , 2

##### 23.1-2 GOuterGroupHomomorphismNC
 ‣ GOuterGroupHomomorphismNC ( global variable )
 ‣ GOuterGroupHomomorphismNC ( global variable )

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.

##### 23.1-3 GOuterHomomorphismTester
 ‣ GOuterHomomorphismTester( A, B, phi ) ( function )

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.

##### 23.1-4 Centre
 ‣ Centre( A ) ( function )

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

1 , 2 , 3 , 4

##### 23.1-5 DirectProductGog
 ‣ DirectProductGog( A, B ) ( function )
 ‣ DirectProductGog( Lst ) ( function )

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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