<Chapter><Heading> G-Outer Groups</Heading> <Section><Heading> &nbsp;</Heading> 
<ManSection> <Func Name="GOuterGroup" Arg="E,N"/> <Func Name="GOuterGroup" Arg=""/> <Description> <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). <P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap7.html</Link><LinkText>2</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutCoefficientSequence.html</Link><LinkText>3</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutGouter.html</Link><LinkText>4</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Var Name="GOuterGroupHomomorphismNC"/> <Var Name="GOuterGroupHomomorphismNC"/> <Description> <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. <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="GOuterHomomorphismTester" Arg="A,B,phi"/> <Description> <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. <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="Centre" Arg="A"/> <Description> <P/> Inputs G-outer group <M>A</M> and returns the group theoretic centre of ActedGroup(A) as a G-outer group. <P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap7.html</Link><LinkText>2</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutParallel.html</Link><LinkText>3</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutSchurMultiplier.html</Link><LinkText>4</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutGouter.html</Link><LinkText>5</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutLieCovers.html</Link><LinkText>6</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="DirectProductGog" Arg="A,B"/> <Func Name="DirectProductGog" Arg="Lst"/> <Description> <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). <P/><B>Examples:</B> 
</Description> </ManSection> </Section> </Chapter>