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

<Table Align="|l|" >

<Row>
<Item>
<Index>AutomorphismGroupAsCatOneGroup</Index>
<C>AutomorphismGroupAsCatOneGroup(G)</C>


<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>.
</Item>
</Row>

<Row>
<Item>
<Index>HomotopyGroup</Index>
<C>HomotopyGroup(C,n)</C>


<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>. 
 
</Item>
</Row>

<Row>
<Item>
<Index>HomotopyModule</Index>
<C>HomotopyModule(C,2)</C>


<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.

</Item>
</Row>

<Row>
<Item>
<Index>QuasiIsomorph</Index>
<C>QuasiIsomorph(C)</C>


<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>. 

</Item>
</Row>


<Row>
<Item>
<Index>ModuleAsCatOneGroup</Index>
<C>ModuleAsCatOneGroup(G,alpha,M)</C>


<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>.
</Item>
</Row>

<Row>
<Item>
<Index>MooreComplex</Index>
<C>MooreComplex(C)</C>


<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).
</Item>
</Row>

<Row>
<Item>
<Index>NormalSubgroupAsCatOneGroup</Index>
<C>NormalSubgroupAsCatOneGroup(G,N)</C>


<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>.

</Item>
</Row>

<Row>
<Item>
<Index>XmodToHAP</Index>
<C>XmodToHAP(C)</C>


<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.  
</Item>
</Row>


</Table>
</Chapter>


