<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 th 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>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 <M>[M_1
\rightarrow M_0]</M> as a list whose single entry is a homomorphism
of 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>


</Table>
</Chapter>


