In <Package>HAP</Package> a cat<M>^1</M>-group is called a
<C>CatOneGroup</C> and the traditional terms
<E>source</E> and <E>target</E> are used for the
<C>TailMap</C> and <C>HeadMap</C>.
A <C>CatOneGroup</C> is a record <C>C</C> with fields
<C>C!.sourceMap</C> and <C>C!.targetMap</C>.
<ManSection>
<Oper Name="SmallCat1Group"
Arg="n i j" />
<Description>
This operation calls the <Package>HAP</Package> function
<C>SmallCatOneGroup(n,i,j)</C> which returns a <C>CatOneGroup</C>
from the <Package>HAP</Package> database.
This is then converted into an <Package>XMod</Package> cat<M>^1</M>-group.
Note that the numbering is not the same as that used by the
<Package>XMod</Package> operation <C>Cat1Select</C>.
In the example <C>C12</C> is the converted form of <C>H12</C>.
</Description>
</ManSection>
<P/>
<Example>
<![CDATA[
gap> H12 := SmallCatOneGroup( 12, 4, 3 );
Cat-1-group with underlying group Group( [ f1, f2, f3 ] ) .
gap> C12 := SmallCat1Group( 12, 4, 3 );
[Group( [ f1, f2, f3 ] )=>Group( [ f1, f2, <identity> of ... ] )]
]]>
</Example>
<Description>
These two functions convert between the two alternative implementations.
</Description>
</ManSection>
<P/>
<Example>
<![CDATA[
gap> C12 := CatOneGroupToXMod( H12 );
[Group( [ f1, f2, f3 ] )=>Group( [ f1, f2, <identity> of ... ] )]
gap> C18 := Cat1Select( 18, 4, 3 );
[(C3 x C3) : C2=>Group( [ f1, <identity> of ..., f3 ] )]
gap> H18 := Cat1GroupToHAP( C18 );
Cat-1-group with underlying group (C3 x C3) : C2 .
]]>
</Example>
<ManSection>
<Oper Name="IdCat1Group"
Arg="C" />
<Description>
This function calls the <Package>HAP</Package> function <C>IdCatOneGroup</C>
on a cat<M>^1</M>-group <M>C</M>.
This returns <M>[n,i,j]</M> if the cat<M>^1</M>-group is the <M>j</M>-th
structure on the <C>SmallGroup(n,i)</C>.
</Description>
</ManSection>
<P/>
<Example>
<![CDATA[
gap> IdCatOneGroup( H18 );
[ 18, 4, 4 ]
gap> IdCat1Group( C18 );
[ 18, 4, 4 ]
]]>
</Example>
</Section>
</Chapter>
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