<Chapter><Heading> G-Outer Groups</Heading>

<Table Align="|l|" >

<Row>
<Item>
<Index>GOuterGroup</Index>
<C>GOuterGroup(E,N)</C>
<Br/>
<C>GOuterGroup()</C>
<P/>

Inputs a group <M>E</M> and normal subgroup <M>N</M>. It returns <M>N</M>
as a <M>G</M>-outer group where <M>G=E/N</M>.

<P/>
The function can be used without an argument. In this case an empty outer group <M>C</M> is returned. The components must be set using SetActingGroup(C,G),
SetActedGroup(C,N) and SetOuterAction(C,alpha).
 
</Item>
</Row>

<Row>
<Item>
<Index>GOuterGroupHomomorphismNC</Index>
<C>GOuterGroupHomomorphismNC(A,B,phi)</C>
<Br/>
<C>GOuterGroupHomomorphismNC()</C>
<P/>

Inputs  G-outer groups <M>A</M> and <M>B</M> 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.


<P/>
The function can be used without an argument. In this case an empty outer group homomorphism <M>PHI</M> is returned. The components must then be set.

</Item>
</Row>

<Row>
<Item>
<Index>GOuterHomomorphismTester</Index>
<C>GOuterHomomorphismTester(A,B,phi)</C>
<P/>

Inputs  G-outer groups <M>A</M> and <M>B</M> 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.


<P/>
The function can be used without an argument. In this case an empty outer group homomorphism <M>PHI</M> is returned. The components must then be set. 
</Item>
</Row>


<Row>
<Item>
<Index>Centre</Index>
<C>Centre(A)</C>
<P/>

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

</Item>
</Row>

<Row>
<Item>
<Index>DirectProductGog</Index>
<C>DirectProductGog(A,B)</C>
<Br/>
<C>DirectProductGog(Lst)</C>
<P/>

Inputs  G-outer groups <M>A</M> and <M>B</M> 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.

<P/>
The function also applies to a list Lst of G-outer groups  with common acting group.

<P/>
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).
</Item>
</Row>



</Table>
</Chapter>