Quelle isoclinic.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>

<Chapter Label="chap-isclnc">
<Heading>Isoclinism of groups and crossed modules</Heading>

This chapter describes some functions written by Alper Odabaş and
Enver Uslu, and reported in their paper <Cite Key="IOU1" />.
Section <Ref Sect="sect-more-xmod-ops" /> contains some additional
basic functions for crossed modules,
constructing quotients, centres, centralizers and normalizers.
In Sections <Ref Sect="sect-isoclinic-groups" /> and
<Ref Sect="sect-isoclinic-xmods" />
there are functions dealing specifically with isoclinism
for groups and for crossed modules.
Since these functions represent a recent addition to the package
(as of November 2015),
the function names are liable to change in future versions.
The notion of isoclinism has been crucial to the enumeration of groups
of prime power order, see for example James, Newman and O'Brien,
<Cite Key="JNO" />.

<Section Label="sect-more-xmod-ops">
<Heading>More operations for crossed modules</Heading>

<ManSection>
 <Oper Name="FactorPreXMod"
 Arg="X1 X2" />
 <Oper Name="NaturalMorphismByNormalSubPreXMod"
 Arg="X1 X2" />
<Description>
When <M>\calX_2 = (\partial_2 : S_2 \to R_2)</M>
is a normal sub-precrossed module of
<M>\calX_1 = (\partial_1 : S_1 \to R_1)</M>,
then the quotient precrossed module is
<M>(\partial : S_2/S_1 \to R_2/R_1)</M>
with the induced boundary and action maps.
Quotienting a precrossed module by it's Peiffer subgroup
is a special case of this construction.
(Permutation representations vary between different versions of &GAP;.)
</Description>
</ManSection>

<Example>
<![CDATA[
gap> d24 := DihedralGroup( IsPermGroup, 24 );;
gap> SetName( d24, "d24" );
gap> Y24 := XModByAutomorphismGroup( d24 );;
gap> Size2d( Y24 );
[ 24, 48 ]
gap> X24 := Image( IsomorphismPerm2DimensionalGroup( Y24 ) );
[d24->Group([ (2,4), (1,2,3,4), (6,7), (5,6,7) ])]
gap> nsx := NormalSubXMods( X24 );;
gap> Length( nsx );
40
gap> ids := List( nsx, n -> IdGroup(n) );;
gap> pos1 := Position( ids, [ [4,1], [8,3] ] );;
gap> Xn1 := nsx[pos1];
[Group( [ f2*f4^2, f3*f4 ] )->Group( [ f3, f4, f5 ] )]
gap> nat1 := NaturalMorphismByNormalSubPreXMod( X24, Xn1 );;
gap> Qn1 := FactorPreXMod( X24, Xn1 );;
gap> [ Size2d( Xn1 ), Size2d( Qn1 ) ];
[ [ 4, 8 ], [ 6, 6 ] ]
]]>
</Example>

<ManSection>
 <Oper Name="IntersectionSubXMods"
 Arg="X0 X1 X2" />
<Description>
When <C>X1,X2</C> are subcrossed modules of <C>X0</C>,
then the source and range of their intersection are the intersections
of the sources and ranges of <C>X1</C> and <C>X2</C> respectively.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> pos2 := Position( ids, [ [24,6], [12,4] ] );;
gap> Xn2 := nsx[pos2];;
gap> IdGroup( Xn2 );
[ [ 24, 6 ], [ 12, 4 ] ]
gap> pos3 := Position( ids, [ [12,2], [24,5] ] );;
gap> Xn3 := nsx[pos3];;
gap> IdGroup( Xn3 );
[ [ 12, 2 ], [ 24, 5 ] ]
gap> Xn23 := IntersectionSubXMods( X24, Xn2, Xn3 );;
gap> IdGroup( Xn23 );
[ [ 12, 2 ], [ 6, 2 ] ]
]]>
</Example>

<ManSection>
 <Oper Name="Displacement"
 Arg="alpha r s" />
 <Oper Name="DisplacementGroup"
 Arg="X0 Q T" />
 <Attr Name="DisplacementSubgroup"
 Arg="X0" />
<Description>
Commutators may be written
<M>[r,q] = r^{-1}q^{-1}rq = (q^{-1})^rq = r^{-1}r^q</M>,
and satisfy identities
<Display>
[r,q]^p = [r^p,q^p], \qquad
[pr,q] = [p,q]^r[r,q], \qquad
[r,pq] = [r,q][r,p]^q, \qquad
[r,q]^{-1} = [q,r].
</Display>
In a similar way, when a group <M>R</M> acts on a group <M>S</M>,
the <E>displacement</E> of <M>s \in S</M> by <M>r \in R</M>
is defined to be <M>\langle r,s \rangle := (s^{-1})^rs \in S</M>.
When <M>\calX = (\partial : S \to R)</M> is a pre-crossed module,
the first crossed module axiom requires
<M>\partial\langle r,s \rangle = [r,\partial s]</M>.
When <M>\alpha</M> is the action of <M>\calX</M>,
the <Ref Oper="Displacement"/> function may be used to calculate
<M>\langle r,s \rangle</M>.
Displacements satisfy the following identities,
where <M>s,t \in S,~ p,q,r \in R</M>:
<Display>
\langle r,s \rangle^p = \langle r^p,s^p \rangle, \qquad
\langle qr,s \rangle = \langle q,s \rangle^r \langle r,s \rangle, \qquad
\langle r,st \rangle = \langle r,t \rangle \langle r,s \rangle^t, \qquad
\langle r,s \rangle^{-1} = \langle r^{-1},s^r \rangle.
</Display>
The operation <Ref Oper="DisplacementGroup"/> applied to <C>X0,Q,T</C>
is the subgroup of <C>S</C> consisting of all the displacements
<M>\langle r,s \rangle, r \in Q \leqslant R, s \in T \leqslant S</M>.
The <Ref Attr="DisplacementSubgroup"/> of <M>\calX</M> is the subgroup
<M>\Disp(\calX)</M> of <M>S</M>
given by <C>DisplacementGroup(X0,R,S)</C>.
The identities imply
<M>\langle r,s \rangle^t =
 \langle r,st^{r^{-1}} \rangle \langle r^{-1},t \rangle</M>,
so <M>\Disp(\calX)</M> is normal in <M>S</M>.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> pos4 := Position( ids, [ [6,2], [24,14] ] );;
gap> Xn4 := nsx[pos4];;
gap> bn4 := Boundary( Xn4 );;
gap> Sn4 := Source(Xn4);;
gap> Rn4 := Range(Xn4);;
gap> genRn4 := GeneratorsOfGroup( Rn4 );;
gap> L := List( genRn4, g -> ( Order(g) = 2 ) and
> not ( IsNormal( Rn4, Subgroup( Rn4, [g] ) ) ) );;
gap> pos := Position( L, true );;
gap> s := Sn4.1; r := genRn4[pos];
(1,3,5,7,9,11)(2,4,6,8,10,12)
(6,7)
gap> act := XModAction( Xn4 );;
gap> d := Displacement( act, r, s );
(1,5,9)(2,6,10)(3,7,11)(4,8,12)
gap> Image( bn4, d ) = Comm( r, Image( bn4, s ) );
true
gap> Qn4 := Subgroup( Rn4, [ (6,7), (1,3), (2,4) ] );;
gap> Tn4 := Subgroup( Sn4, [ (1,3,5,7,9,11)(2,4,6,8,10,12) ] );;
gap> DisplacementGroup( Xn4, Qn4, Tn4 );
Group([ (1,5,9)(2,6,10)(3,7,11)(4,8,12) ])
gap> DisplacementSubgroup( Xn4 );
Group([ (1,5,9)(2,6,10)(3,7,11)(4,8,12) ])
]]>
</Example>

<ManSection>
 <Oper Name="CommutatorSubXMod"
 Arg="X X1 X2" />
 <Oper Name="CrossActionSubgroup"
 Arg="X X1 X2" />
<Description>
When <M>\calX_1 = (N \to Q), \calX_2 = (M \to P)</M>
are two normal subcrossed modules of <M>\calX = (\partial : S \to R)</M>,
the displacements <M>\langle p,n \rangle</M> and <M>\langle q,m \rangle</M>
all map by <M>\partial</M> into <M>[Q,P]</M>.
These displacements form a normal subgroup of <M>S</M>,
called the <C>CrossActionSubgroup</C>.
The <C>CommutatorSubXMod</C> <M>[\calX_1,\calX_2]</M>
has this subgroup as source and <M>[P,Q]</M> as range,
and is normal in <M>\calX</M>.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> CAn23 := CrossActionSubgroup( X24, Xn2, Xn3 );;
gap> IdGroup( CAn23 );
[ 12, 2 ]
gap> Cn23 := CommutatorSubXMod( X24, Xn2, Xn3 );;
gap> IdGroup( Cn23 );
[ [ 12, 2 ], [ 6, 2 ] ]
gap> Xn23 = Cn23;
true
]]>
</Example>

<ManSection>
 <Attr Name="DerivedSubXMod"
 Arg="X0" />
<Description>
The <C>DerivedSubXMod</C> of <M>\calX</M> is the normal subcrossed module
<M>[\calX,\calX] = (\partial' : \Disp(\calX) \to [R,R])
where <M>\partial' is the restriction of \partial
(see page 66 of Norrie's thesis ).
</Description>
</ManSection>

<Example>
<![CDATA[
gap> DXn4 := DerivedSubXMod( Xn4 );;
gap> IdGroup( DXn4 );
[ [ 3, 1 ], [ 3, 1 ] ]
]]>
</Example>

<ManSection>
 <Oper Name="FixedPointSubgroupXMod"
 Arg="X0 T Q" />
 <Oper Name="StabilizerSubgroupXMod"
 Arg="X0 T Q" />
<Description>
The <C>FixedPointSubgroupXMod(X,T,Q)</C>
for <M>\calX=(\partial : S \to R)</M> is the subgroup
<M>\Fix(\calX,T,Q)</M>
of elements <M>t \in T \leqslant S</M> individually fixed
under the action of <M>Q \leqslant R</M>.

The <C>StabilizerSubgroupXMod(X,T,Q)</C> for <M>\calX</M>
is the subgroup <M>\Stab(\calX,T,Q)</M> of <M>Q \leqslant R</M>
whose elements act trivially on the whole of <M>T \leqslant S</M>
(see page 19 of Norrie's thesis ).
</Description>
</ManSection>

<Example>
<![CDATA[
gap> fix := FixedPointSubgroupXMod( Xn4, Sn4, Rn4 );
Group([ (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) ])
gap> stab := StabilizerSubgroupXMod( Xn4, Sn4, Rn4 );;
gap> IdGroup( stab );
[ 12, 5 ]
]]>
</Example>

<ManSection>
 <Attr Name="CentreXMod"
 Arg="X0" />
 <Oper Name="Centralizer"
 Arg="X Y" />
 <Oper Name="Normalizer"
 Arg="X Y" />
<Description>
The <E>centre</E> <M>Z(\calX)</M>
of <M>\calX = (\partial : S \to R)</M>
has as source the fixed point subgroup <M>\Fix(\calX,S,R)</M>.
The range is the intersection of the centre <M>Z(R)</M>
with the stabilizer subgroup.

When <M>\calY = (T \to Q)</M> is a subcrossed module of
<M>\calX = (\partial : S \to R)</M>, the <E>centralizer</E>
<M>C_{\calX}(\calY)</M> of <M>\calY</M>
has as source the fixed point subgroup <M>\Fix(\calX,S,Q)</M>.
The range is the intersection of the centralizer <M>C_R(Q)</M>
with <M>\Stab(\calX,T,R)</M>.

The <E>normalizer</E> <M>N_{\calX}(\calY)</M>
of <M>\calY</M> has as source the subgroup of <M>S</M>
consisting of the displacements <M>\langle s,q \rangle</M>
which lie in <M>S</M>.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> ZXn4 := CentreXMod( Xn4 );
[Group( [ f3*f4 ] )->Group( [ f3, f5 ] )]
gap> IdGroup( ZXn4 );
[ [ 2, 1 ], [ 4, 2 ] ]
gap> CDXn4 := Centralizer( Xn4, DXn4 );
[Group( [ f3*f4 ] )->Group( [ f2 ] )]
gap> IdGroup( CDXn4 );
[ [ 2, 1 ], [ 3, 1 ] ]
gap> NDXn4 := Normalizer( Xn4, DXn4 );
[Group( <identity> of ... )->Group( [ f5, f2*f3 ] )]
gap> IdGroup( NDXn4 );
[ [ 1, 1 ], [ 12, 5 ] ]
]]>
</Example>

<ManSection>
 <Attr Name="CentralQuotient"
 Arg="G" />
<Description>
The <C>CentralQuotient</C> of a group <M>G</M> is the crossed module
<M>(G \to G/Z(G))</M>
with the natural homomorphism as the boundary map.
This is a special case of <Ref Oper="XModByCentralExtension"/>.

Similarly, the central quotient of a crossed module
<M>\calX</M> is the crossed square
<M>(\calX \Rightarrow \calX/Z(\calX))</M>
(see section <Ref Sect="sect-xsq-constructions"/>).
</Description>
</ManSection>

<Example>
<![CDATA[
gap> Q24 := CentralQuotient( d24); IdGroup( Q24 );
[d24->d24/Z(d24)]
[ [ 24, 6 ], [ 12, 4 ] ]
]]>
</Example>

<ManSection>
 <Prop Name="IsAbelian2DimensionalGroup"
 Arg="X0" />
 <Prop Name="IsAspherical2DimensionalGroup"
 Arg="X0" />
 <Prop Name="IsSimplyConnected2DimensionalGroup"
 Arg="X0" />
 <Prop Name="IsFaithful2DimensionalGroup"
 Arg="X0" />
<Description>
A crossed module is <E>abelian</E> if it equal to its centre.
This is the case when the range group is abelian and the action is trivial.

A crossed module is <E>aspherical</E> if the boundary has trivial kernel.

A crossed module is <E>simply connected</E> if the boundary
has trivial cokernel.

A crossed module is <E>faithful</E> if the action is faithful.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> [ IsAbelian2DimensionalGroup(Xn4), IsAbelian2DimensionalGroup(X24) ];
[ false, false ]
gap> pos7 := Position( ids, [ [3,1], [6,1] ] );;
gap> [ IsAspherical2DimensionalGroup(nsx[pos7]), IsAspherical2DimensionalGroup(X24) ];
[ true, false ]
gap> [ IsSimplyConnected2DimensionalGroup(Xn4), IsSimplyConnected2DimensionalGroup(X24) ];
[ true, true ]
gap> [ IsFaithful2DimensionalGroup(Xn4), IsFaithful2DimensionalGroup(X24) ];
[ false, true ]
]]>
</Example>

<ManSection>
 <Attr Name="LowerCentralSeriesOfXMod"
 Arg="X0" />
 <Prop Name="IsNilpotent2DimensionalGroup"
 Arg="X0" />
 <Attr Name="NilpotencyClass2DimensionalGroup"
 Arg="X0" />
<Description>
Let <M>\calY</M> be a subcrossed module of <M>\calX</M>.
A <E> series of length</E> <M>n</M> from <M>\calX</M>
to <M>\calY</M> has the form
<Display>
\calX ~=~ \calX_0 ~\unrhd~ \calX_1 ~\unrhd~ \cdots
 ~\unrhd~ \calX_i ~\unrhd~ \cdots
 ~\unrhd~ \calX_n ~=~ \calY
 \quad (1 \leqslant i \leqslant n).
</Display>
The quotients <M>\calF_i = \calX_i / \calX_{i-1}</M>
are the <E>factors</E> of the series.

A factor <M>\calF_i</M> is <E>central</E> if
<M>\calX_{i-1} \unlhd \calX</M> and
<M>\calF_i</M> is a subcrossed module of the centre of
<M>\calX / \calX_{i-1}</M>.

A series is <E>central</E> if all its factors are central.

<M>\calX</M> is <E>soluble</E> if it has a series
all of whose factors are abelian.

<M>\calX</M> is <E>nilpotent</E> is it has a series
all of whose factors are central factors of <M>\calX</M>.

The <E>lower central series</E> of <M>\calX</M>
is the sequence (see <Cite Key="N2" />, p.77):
<Display>
\calX ~=~ \Gamma_1(\calX) ~\unrhd~ \Gamma_2(\calX)
 ~\unrhd~ \cdots
\qquad \mbox{where} \qquad
\Gamma_j(\calX) ~=~ [ \Gamma_{j-1}(\calX), \calX].
</Display>
If <M>\calX</M> is nilpotent, then its lower central series
is its most rapidly descending central series.

The least integer <M>c</M> such that <M>\Gamma_{c+1}(\calX)</M>
is the trivial crossed module is the <E>nilpotency class</E>
of <M>\calX</M>.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> lcs := LowerCentralSeries( X24 );;
gap> List( lcs, g -> IdGroup(g) );
[ [ [ 24, 6 ], [ 48, 38 ] ], [ [ 12, 2 ], [ 6, 2 ] ], [ [ 6, 2 ], [ 3, 1 ] ],
 [ [ 3, 1 ], [ 3, 1 ] ] ]
gap> IsNilpotent2DimensionalGroup( X24 );
false
gap> NilpotencyClassOf2DimensionalGroup( X24 );
0
]]>
</Example>

<ManSection>
 <Oper Name="IsomorphismXMods"
 Arg="X1 X2" />
<Description>
The function <C>IsomorphismXMods</C> computes an isomorphism
<M>\mu : \calX_1 \to \calX_2</M>,
provided one exists, or else returns <C>fail</C>.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> gend24 := GeneratorsOfGroup( d24 );;
gap> a := gend24[1];; b:= gend24[2];;
gap> J := Subgroup( d24, [a^2,b] );
Group([ (1,3,5,7,9,11)(2,4,6,8,10,12), (2,12)(3,11)(4,10)(5,9)(6,8) ])
gap> K := Subgroup( d24, [a^2,a*b] );
Group([ (1,3,5,7,9,11)(2,4,6,8,10,12), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7) ])
gap> XJ := XModByNormalSubgroup( d24, J );;
gap> XK := XModByNormalSubgroup( d24, K );;
gap> iso := IsomorphismXMods( XJ, XK );;
gap> SourceHom( iso );
[ (1,3,5,7,9,11)(2,4,6,8,10,12), (2,12)(3,11)(4,10)(5,9)(6,8) ] ->
[ (1,3,5,7,9,11)(2,4,6,8,10,12), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7) ]
gap> RangeHom( iso );
[ (1,2,3,4,5,6,7,8,9,10,11,12), (2,12)(3,11)(4,10)(5,9)(6,8) ] ->
[ (1,2,3,4,5,6,7,8,9,10,11,12), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7) ]
]]>
</Example>

<ManSection>
 <Func Name="AllXMods"
 Arg="args" />
 <Oper Name="AllXModsWithGroups"
 Arg="src rng" />
 <Oper Name="AllXModsUpToIsomorphism"
 Arg="src rng" />
 <Oper Name="IsomorphismClassRepresentatives2dGroups"
 Arg="L" />
<Description>
The global function <C>AllXMods</C> may be called in three ways.
Firstly, as <C>AllXMods(S,R)</C> to compute all crossed modules with chosen
source and range groups: this calls <C>AllXModsWithGroups(S,R)</C>.
Secondly, <C>AllXMods([n,m])</C> computes all crossed modules with a
given size <C>[n,m]</C>.
Thirdly <C>AllXMods(ord)</C> to compute all crossed modules whose associated
cat1-groups have a given size <C>ord</C>.

The function <C>AllXModsUpToIsomorphism(S,R)</C> returns a list of
representatives of the isomorphism classes of crossed modules
with source <M>S</M> and range <M>R</M>.

If <C>L</C> is a list returned by, for example, <C>AllXModsWithGroups(S,R)</C>,
then the isomorphism class representatives for this list is returned by
<C>IsomorphismClassRepresentatives2dGroups(L)</C>. This result is the same
as that given by <C>AllXModsUpToIsomorphism(S,R)</C>.

In the example we see that there are <M>4</M> crossed modules,
in <M>3</M> isomorphism classes,
<M>(C_6 \to S_3)</M>; forming a subset of the <M>17</M> crossed modules
with size <C>[6,6]</C>; and that these form a subset of the <M>205</M>
crossed modules whose cat1-group has size <M>36</M>.

There are <M>40</M> precrossed modules with size <C>[6,6]</C>.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> c6 := SmallGroup( 6, 2 );;
gap> s3 := SmallGroup( 6, 1 );;
gap> Ac6s3 := AllXMods( c6, s3 );;
gap> Length( Ac6s3 );
4
gap> Ic6s3 := AllXModsUpToIsomorphism( c6, s3 );;
gap> List( Ic6s3, obj -> IsTrivialAction2DimensionalGroup( obj ) );
[ true, false, false ]
gap> Kc6s3 := List( Ic6s3, obj -> KernelCokernelXMod( obj ) );;
gap> List( Kc6s3, obj -> IdGroup( obj ) );
[ [ [ 6, 2 ], [ 6, 1 ] ], [ [ 6, 2 ], [ 6, 1 ] ], [ [ 2, 1 ], [ 2, 1 ] ] ]
[ ]
gap> A66 := AllXMods( [6,6] );;
gap> Length( A66 );
17
gap> IA66 := IsomorphismClassRepresentatives2dGroups( A66 );;
gap> Length( IA66 );
9
gap> x36 := AllXMods( 36 );;
gap> Length( x36 );
205
gap> size36 := List( x36, x -> Size2d( x ) );;
gap> Collected( size36 );
[ [ [ 1, 36 ], 14 ], [ [ 2, 18 ], 7 ], [ [ 3, 12 ], 21 ], [ [ 4, 9 ], 14 ],
 [ [ 6, 6 ], 17 ], [ [ 9, 4 ], 102 ], [ [ 12, 3 ], 8 ], [ [ 18, 2 ], 18 ],
 [ [ 36, 1 ], 4 ] ]
]]>
</Example>

</Section>

<Section Label="sect-isoclinic-groups">
<Heading>Isoclinism for groups</Heading>

<ManSection>
 <Oper Name="Isoclinism"
 Arg="G H" Label="for groups" />
 <Oper Name="AreIsoclinicDomains"
 Arg="G H" Label="for groups" />
<Description>
Let <M>G,H</M> be groups with central quotients <M>Q(G)</M> and <M>Q(H)</M>
and derived subgroups <M>[G,G]</M> and <M>[H,H]</M> respectively.
Let <M>c_G : G/Z(G) \times G/Z(G) \to [G,G]</M>
and <M>c_H : H/Z(H) \times H/Z(H) \to [H,H]</M> be the two commutator maps.
An <E>isoclinism</E> <M>G \sim H</M> is a pair of isomorphisms
<M>(\eta,\xi)</M> where <M>\eta : G/Z(G) \to H/Z(H)</M> and
<M>\xi : [G,G] \to [H,H]</M> such that
<M>c_G * \xi = (\eta \times \eta) * c_H</M>.
Isoclinism is an equivalence relation, and all abelian groups are isoclinic
to the trivial group.

</Description>
</ManSection>

<Example>
<![CDATA[
gap> G := SmallGroup( 64, 6 );; StructureDescription( G );
"(C8 x C4) : C2"
gap> QG := CentralQuotient( G );; IdGroup( QG );
[ [ 64, 6 ], [ 8, 3 ] ]
gap> H := SmallGroup( 32, 41 );; StructureDescription( H );
"C2 x Q16"
gap> QH := CentralQuotient( H );; IdGroup( QH );
[ [ 32, 41 ], [ 8, 3 ] ]
gap> Isoclinism( G, H );
[ [ f1, f2, f3 ] -> [ f1, f2*f3, f3 ], [ f3, f5 ] -> [ f4*f5, f5 ] ]
gap> K := SmallGroup( 32, 43 );; StructureDescription( K );
"(C2 x D8) : C2"
gap> QK := CentralQuotient( K );; IdGroup( QK );
[ [ 32, 43 ], [ 16, 11 ] ]
gap> AreIsoclinicDomains( G, K );
false
]]>
</Example>

<ManSection>
 <Prop Name="IsStemDomain"
 Arg="G" Label="for groups" />
 <Attr Name="IsoclinicStemDomain"
 Arg="G" Label="for groups" />
 <Oper Name="AllStemGroupIds"
 Arg="n" />
 <Oper Name="AllStemGroupFamilies"
 Arg="n" />
<Description>
A group <M>G</M> is a <E>stem group</E> if <M>Z(G) \leq [G,G]</M>.
Every group is isoclinic to a stem group, but distinct stem groups
may be isoclinic.
For example, groups <M>D_8, Q_8</M> are two isoclinic stem groups.

The function <C>IsoclinicStemDomain </C> returns a stem group
isoclinic to <M>G</M>.

The function <C>AllStemGroupIds</C> returns the <C>IdGroup</C> list
of the stem groups of a specified size, while <C>AllStemGroupFamilies</C>
splits this list into isoclinism classes.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> DerivedSubgroup(G);
Group([ f3, f5 ])
gap> IsStemDomain( G );
false
gap> IsoclinicStemDomain( G );
<pc group of size 16 with 4 generators>
gap> AllStemGroupIds( 32 );
[ [ 32, 6 ], [ 32, 7 ], [ 32, 8 ], [ 32, 18 ], [ 32, 19 ], [ 32, 20 ],
 [ 32, 27 ], [ 32, 28 ], [ 32, 29 ], [ 32, 30 ], [ 32, 31 ], [ 32, 32 ],
 [ 32, 33 ], [ 32, 34 ], [ 32, 35 ], [ 32, 43 ], [ 32, 44 ], [ 32, 49 ],
 [ 32, 50 ] ]
gap> AllStemGroupFamilies( 32 );
[ [ [ 32, 6 ], [ 32, 7 ], [ 32, 8 ] ], [ [ 32, 18 ], [ 32, 19 ], [ 32, 20 ] ],
 [ [ 32, 27 ], [ 32, 28 ], [ 32, 29 ], [ 32, 30 ], [ 32, 31 ], [ 32, 32 ],
 [ 32, 33 ], [ 32, 34 ], [ 32, 35 ] ], [ [ 32, 43 ], [ 32, 44 ] ],
 [ [ 32, 49 ], [ 32, 50 ] ] ]
]]>
</Example>

<ManSection Label="group-rank">
 <Attr Name="IsoclinicRank"
 Arg="G" Label="for groups" />
 <Attr Name="IsoclinicMiddleLength"
 Arg="G" Label="for groups" />
<Description>
Let <M>G</M> be a finite <M>p</M>-group.
Then <M>\log_p |[G,G] / (Z(G) \cap [G,G])|</M> is called the
<E>middle length</E> of <M>G</M>.
Also <M>\log_p |Z(G) \cap [G,G]| + \log_p |G/Z(G)|</M>
is called the <E>rank</E> of <M>G</M>.
These invariants appear in the tables of isoclinism families of
groups of order 128 in <Cite Key="JNO" />.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> IsoclinicMiddleLength( G );
1
gap> IsoclinicRank( G );
4
]]>
</Example>

</Section>

<Section Label="sect-isoclinic-xmods">
<Heading>Isoclinism for crossed modules</Heading>

<ManSection>
 <Oper Name="Isoclinism"
 Arg="X0 Y0" Label="for crossed modules" />
 <Oper Name="AreIsoclinicDomains"
 Arg="X0 Y0" Label="for crossed modules of groups" />
<Description>
Let <M>\calX,\calY</M> be crossed modules with central quotients
<M>Q(\calX)</M> and <M> Q(\calY)</M>,
and derived subcrossed modules <M>[\calX,\calX]</M>
and <M>[\calY,\calY]</M> respectively.
Let <M>c_\calX : Q(\calX) \times Q(\calX)
 \to [\calX,\calX]</M>
and <M>c_\calY : Q(\calY) \times Q(\calY)
 \to [\calY,\calY]</M>
be the two commutator maps.
An <E>isoclinism</E> <M>\calX \sim \calY</M>
is a pair of bijective morphisms
<M>(\eta,\xi)</M> where <M>\eta : Q(\calX) \to Q(\calY)</M>
and <M>\xi : [\calX,\calX] \to [\calY,\calY]</M>
such that
<M>c_\calX * \xi = (\eta \times \eta) * c_\calY</M>.
Isoclinism is an equivalence relation, and all abelian crossed modules
are isoclinic to the trivial crossed module.

</Description>
</ManSection>

<Example>
<![CDATA[
gap> C8 := Cat1Group( 16, 8, 2 );;
gap> X8 := XMod(C8); IdGroup( X8 );
[Group( [ f1*f2*f3, f3, f4 ] )->Group( [ f2, f2 ] )]
[ [ 8, 1 ], [ 2, 1 ] ]
gap> C9 := Cat1Group( 32, 9, 2 );
[(C8 x C2) : C2 => Group( [ f2, f2 ] )]
gap> X9 := XMod( C9 ); IdGroup( X9 );
[Group( [ f1*f2*f3, f3, f4, f5 ] )->Group( [ f2, f2 ] )]
[ [ 16, 5 ], [ 2, 1 ] ]
gap> AreIsoclinicDomains( X8, X9 );
true
gap> ism89 := Isoclinism( X8, X9 );;
gap> Display( ism89 );
[ [[Group( [ f1, f2, <identity> of ... ] ) -> Group( [ f2, f2 ] )] => [Group(
 [ f1, f2, <identity> of ..., <identity> of ... ] ) -> Group(
 [ f2, f2 ] )]],
 [[Group( [ f3 ] ) -> Group( <identity> of ... )] => [Group(
 [ f3 ] ) -> Group( <identity> of ... )]] ]
]]>
</Example>

<ManSection>
 <Prop Name="IsStemDomain"
 Arg="X0" Label="for crossed modules of groups" />
 <Attr Name="IsoclinicStemDomain"
 Arg="X0" Label="for crossed modules of groups" />
<Description>
A crossed module <M>\calX</M> is a <E>stem crossed module</E>
if <M>Z(\calX) \leq [\calX,\calX]</M>.
Every crossed module is isoclinic to a stem crossed module,
but distinct stem crossed modules may be isoclinic.

A method for <C>IsoclinicStemDomain</C> has yet to be implemented.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> IsStemDomain(X8);
true
gap> IsStemDomain(X9);
false
]]>
</Example>

<ManSection>
 <Attr Name="IsoclinicRank"
 Arg="X0" Label="for crossed modules of groups" />
 <Attr Name="IsoclinicMiddleLength"
 Arg="X0" Label="for crossed modules of groups" />
<Description>
The formulae in subsection <Ref Subsect="group-rank" />
are applied to the crossed module.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> IsoclinicMiddleLength(X8);
[ 1, 0 ]
gap> IsoclinicRank(X8);
[ 3, 1 ]
]]>
</Example>

</Section>

</Chapter>

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