<Chapter><Heading> Cat-1-groups</Heading>

<Section><Heading> &nbsp;</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>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCrossedMods.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutquasi.html</Link><LinkText>3</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>4</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>5</LinkText></URL>&nbsp; 

</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>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutNonabelian.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCrossedMods.html</Link><LinkText>3</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutquasi.html</Link><LinkText>4</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>5</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>6</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutTensorSquare.html</Link><LinkText>7</LinkText></URL>&nbsp; 

</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>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCrossedMods.html</Link><LinkText>2</LinkText></URL>&nbsp; 

</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>../www/SideLinks/About/aboutquasi.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>2</LinkText></URL>&nbsp; 

</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>&nbsp; 

</Description>

























































































































</ManSection>

</Section>
</Chapter>

