<Chapter><Heading> Cat-1-groups</Heading> <Section><Heading> </Heading>
<ManSection> <Func Name="AutomorphismGroupAsCatOneGroup" Arg="G"/> <Description> <P/> Inputs a group <M>G</M> and returns the Cat-1-group <M>C</M> corresponding to the crossed module <M>G\rightarrow Aut(G)</M>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../tutorial/chap12.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCrossedMods.html</Link><LinkText>3</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutquasi.html</Link><LinkText>4</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>5</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>6</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="HomotopyGroup" Arg="C,n"/> <Description> <P/> Inputs a cat-1-group <M>C</M> and an integer n. It returns the <M>n</M>th homotopy group of <M>C</M>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../tutorial/chap12.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutNonabelian.html</Link><LinkText>3</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCrossedMods.html</Link><LinkText>4</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutquasi.html</Link><LinkText>5</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>6</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>7</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutTensorSquare.html</Link><LinkText>8</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="HomotopyModule" Arg="C,2"/> <Description> <P/> Inputs a cat-1-group <M>C</M> and an integer n=2. It returns the second homotopy group of <M>C</M> as a G-module (i.e. abelian G-outer group) where G is the fundamental group of C. <P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCrossedMods.html</Link><LinkText>2</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="QuasiIsomorph" Arg="C"/> <Description> <P/> Inputs a cat-1-group <M>C</M> and returns a cat-1-group <M>D</M> for which there exists some homomorphism <M>C\rightarrow D</M> that induces isomorphisms on homotopy groups. <P/> This function was implemented by <B>Le Van Luyen</B>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap12.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutquasi.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>3</LinkText></URL>
</Description> </ManSection>
<ManSection> <Var Name="ModuleAsCatOneGroup"/> <Description> <P/> Inputs a group <M>G</M>, an abelian group <M>M</M> and a homomorphism <M>\alpha\colon G\rightarrow Aut(M)</M>. It returns the Cat-1-group <M>C</M> corresponding th the zero crossed module <M>0\colon M\rightarrow G</M>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="MooreComplex" Arg="C"/> <Description> <P/> Inputs a cat-1-group <M>C</M> and returns its Moore complex as a G-complex (i.e. as a complex of groups considered as 1-outer groups). <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="NormalSubgroupAsCatOneGroup" Arg="G,N"/> <Description> <P/> Inputs a group <M>G</M> with normal subgroup <M>N</M>. It returns the Cat-1-group <M>C</M> corresponding th the inclusion crossed module <M> N\rightarrow G</M>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="XmodToHAP" Arg="C"/> <Description> <P/> Inputs a cat-1-group <M>C</M> obtained from the Xmod package and returns a cat-1-group <M>D</M> for which IsHapCatOneGroup(D) returns true. <P/> It returns "fail"id <M>C</M> has not been produced by the Xmod package. <P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutquasi.html</Link><LinkText>1</LinkText></URL>
</Description> </ManSection> </Section> </Chapter>
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