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

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<Table Align="|l|" >

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<Row>

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<Item>

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<Index>GOuterGroup</Index>

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<C>GOuterGroup(E,N)</C>

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<C>GOuterGroup()</C>

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<P/>

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Inputs a group <M>E</M> and normal subgroup <M>N</M>. It returns <M>N</M>

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as a <M>G</M>-outer group where <M>G=E/N</M>.

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<P/>

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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),

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SetActedGroup(C,N) and SetOuterAction(C,alpha).

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</Item>

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</Row>

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<Row>

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<Item>

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<Index>GOuterGroupHomomorphismNC</Index>

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<C>GOuterGroupHomomorphismNC(A,B,phi)</C>

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<C>GOuterGroupHomomorphismNC()</C>

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<P/>

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Inputs  G-outer groups <M>A</M> and <M>B</M> with common acting

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 group, and a group homomorphism phi:ActedGroup(A) --> ActedGroup(B). 

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It returns the corresponding G-outer homomorphism PHI:A--> B.

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 No check is made to verify that phi is actually a group homomorphism which preserves the G-action.

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<P/>

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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.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index>GOuterHomomorphismTester</Index>

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<C>GOuterHomomorphismTester(A,B,phi)</C>

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<P/>

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Inputs  G-outer groups <M>A</M> and <M>B</M> with common acting  group, and a group homomorphism phi:ActedGroup(A) --> ActedGroup(B). 

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 It tests whether  phi is  a group homomorphism which preserves the G-action.

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<P/>

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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. 

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</Item>

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</Row>

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<Row>

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<Item>

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<Index>Centre</Index>

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<C>Centre(A)</C>

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<P/>

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Inputs  G-outer group <M>A</M> and 

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 returns the group theoretic centre of ActedGroup(A) as a

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 G-outer group.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index>DirectProductGog</Index>

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<C>DirectProductGog(A,B)</C>

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<C>DirectProductGog(Lst)</C>

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<P/>

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Inputs  G-outer groups <M>A</M> and <M>B</M> with common acting group,

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and returns their group-theoretic direct product as a G-outer group. 

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The outer action

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on the direct product is the diagonal one.

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<P/>

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

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<P/>

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For a direct product D constructed using this function, the embeddings and projections can be obtained (as G-outer group homomorphisms) using the functions

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 Embedding(D,i) and Projection(D,i).

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</Item>

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</Row>

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</Table>

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</Chapter>

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