##  expt/q8.txt  
##  part 1 : construct permutation groups and homomorphisms: 
p := (1,2,3,4)(5,8,7,6); 
q := (1,5,3,7)(2,6,4,8);
q8 := Group( p, q ); 
SetName( q8, "q8" );
a := p;  b := q;  c := p^2; 
c4a := Subgroup( q8, [a] ); 
SetName( c4a, "c4a" ); 
c4b := Subgroup( q8, [b] ); 
SetName( c4b, "c4b" ); 
c2c := Subgroup( q8, [c] ); 
SetName( c2c, "c2c" ); 
fu := GroupHomomorphismByImages( c2c, c4a, [c], [c] );
fl := GroupHomomorphismByImages( c2c, c4b, [c], [c] );
fd := GroupHomomorphismByImages( c4b, q8, [b], [b] );
fr := GroupHomomorphismByImages( c4a, q8, [a], [a] );

autc4a := AutomorphismGroup( c4a );
imq8a := List( [p,q], g -> ConjugatorAutomorphism( c4a, g ) );
actq8a := GroupHomomorphismByImages( q8, autc4a, [p,q], imq8a );
c4aq8 := SemidirectProduct( q8, actq8a, c4a ); 
SetName( c4aq8, "c4a:q8" );
prc4aq8 := Projection( c4aq8 ); 
Print( "prc4aq8: ", MappingGeneratorsImages(prc4aq8), "\n" ); 
embq8 := Embedding( c4aq8, 1 ); 
p2 := ImageElm( embq8, p );
q2 := ImageElm( embq8, q );
Print( "[p2,q2] = ", [p2,q2], "\n" );
embc4a := Embedding( c4aq8, 2 ); 
a2 := ImageElm( embc4a, a );
Print( "a2 = ", a2, "\n" ); 

autc2c := AutomorphismGroup( c2c );
imc4b := List( [b], g -> ConjugatorAutomorphism( c2c, g ) );
actc4b := GroupHomomorphismByImages( c4b, autc2c, [b], imc4b );
c2c4b := SemidirectProduct( c4b, actc4b, c2c ); 
SetName( c2c4b, "c2c:c4b" ); 
prc2c4b := Projection( c2c4b ); 
Print( "prc2c4b: ", MappingGeneratorsImages(prc2c4b), "\n" ); 
embc4b := Embedding( c2c4b, 1 ); 
b2 := ImageElm( embc4b, b );
Print( "b2 = ", b2, "\n" ); 
embc2c := Embedding( c2c4b, 2 );
c2 := ImageElm( embc2c, c );
Print( "c2 = ", c2, "\n" ); 

## boundary map is the inclusion of c2c4b in c4aq8 
isc2c := ImagesSource( embc2c );
imbdy := [0,0]; 
imbdy[1] := ImageElm( embc4a, ImageElm( fu, c ) ); 
imbdy[2] := ImageElm( embq8, ImageElm( fd, b ) ); 
bdy := GroupHomomorphismByImages( c2c4b, c4aq8, [c2,b2], imbdy ); 
Print( "bdy = ", MappingGeneratorsImages(bdy), "\n" ); 

## action of c4aq8 on c2c4b 
autc2c4b := AutomorphismGroup( c2c4b ); 
imact := [0,0,0]; 
ima := [0,0]; 
ima[1] := ImageElm( embc2c, c^a ); 
ima[2] := b2 * ImageElm( embc2c, Comm(b,a) ); 
imact[1] := GroupHomomorphismByImages( c2c4b, c2c4b, [c2,b2], ima ); 
ima[1] := ImageElm( embc2c, c^p ); 
ima[2] := ImageElm( embc4b, b^p ); 
imact[2] := GroupHomomorphismByImages( c2c4b, c2c4b, [c2,b2], ima ); 
ima[1] := ImageElm( embc2c, c^q ); 
ima[2] := ImageElm( embc4b, b^q ); 
imact[3] := GroupHomomorphismByImages( c2c4b, c2c4b, [c2,b2], ima ); 
if not ForAll( imact, f -> IsBijective(f) ) then 
    Error( "imact homs not all bijective" ); 
fi;
act := GroupHomomorphismByImages( c4aq8, autc2c4b, [a2,p2,q2], imact ); 

G := SemidirectProduct( c4aq8, act, c2c4b ); 
SetName( G, "G" ); 
prG := Projection( G ); 
Print( "prG: ", MappingGeneratorsImages(prG), "\n" ); 
embc4aq8 := Embedding( G, 1 ); 
a3 := ImageElm( embc4aq8, a2 ); 
p3 := ImageElm( embc4aq8, p2 ); 
q3 := ImageElm( embc4aq8, q2 ); 
embc2c4b := Embedding( G, 2 ); 
c3 := ImageElm( embc2c4b, c2 ); 
b3 := ImageElm( embc2c4b, b2 ); 
genG := [ c3, b3, a3, p3, q3 ]; 
Print( "genG = ", genG, "\n" ); 

## cat1-embedding 
e := embc4aq8; 
## tail map 
id := One( c4aq8 ); 
imt := [ id, id, a2, p2, q2 ]; 
t := GroupHomomorphismByImages( G, c4aq8, genG, imt ); 
Print( "t*e*t = t ? ", t*e*t = t, "\n" ); 
## head map 
imh := Concatenation( imbdy, [a2,p2,q2] ); 
Print( "imh = ", imh, "\n" );
## h := GroupHomomorphismByImages( G, c4aq8, genG, imh ); 
h := GroupGeneralMappingByImages( G, c4aq8, genG, imh ); 
ok := IsSingleValued( h ); 
Print( "h is single valued? ", ok, "\n\n" ); 

##  part 2 : construct isomorphic pc-groups and homomorphisms 
Q8 := SmallGroup( 8, 4 ); 
SetName( Q8, "Q8" );
P := Q8.1;
Q := Q8.2;
if not ( ( Order(P) = 4 ) and ( Order(Q) = 4) ) then Error( "orders" ); fi; 
A := P;  B := Q;  C := P^2;
C4A := Subgroup( Q8, [A] ); 
SetName( C4A, "C4A" ); 
C4B := Subgroup( Q8, [B] ); 
SetName( C4B, "C4B" ); 
C2C := Subgroup( Q8, [A^2] ); 
SetName( C2C, "C2C" ); 
FU := GroupHomomorphismByImages( C2C, C4A, [C], [C] );
FL := GroupHomomorphismByImages( C2C, C4B, [C], [C] );
FD := GroupHomomorphismByImages( C4B, Q8, [B], [B] );
FR := GroupHomomorphismByImages( C4A, Q8, [A], [A] );

autC4A := AutomorphismGroup( C4A );
imQ8a := List( [P,Q], g -> ConjugatorAutomorphism( C4A, g ) );
actQ8a := GroupHomomorphismByImages( Q8, autC4A, [P,Q], imQ8a );
C4AQ8 := SemidirectProduct( Q8, actQ8a, C4A ); 
SetName( C4AQ8, "C4A:Q8" );
prC4AQ8 := Projection( C4AQ8 ); 
Print( "prC4AQ8: ", MappingGeneratorsImages(prC4AQ8), "\n" ); 
embQ8 := Embedding( C4AQ8, 1 ); 
P2 := ImageElm( embQ8, P );
Q2 := ImageElm( embQ8, Q );
Print( "[P2,Q2] = ", [P2,Q2], "\n" );
embC4A := Embedding( C4AQ8, 2 ); 
A2 := ImageElm( embC4A, A );
Print( "A2 = ", A2, "\n" ); 

autC2C := AutomorphismGroup( C2C );
imC4B := List( [B], g -> ConjugatorAutomorphism( C2C, g ) );
actC4B := GroupHomomorphismByImages( C4B, autC2C, [B], imC4B );
C2C4B := SemidirectProduct( C4B, actC4B, C2C ); 
SetName( C2C4B, "C2C:C4B" ); 
prC2C4B := Projection( C2C4B ); 
Print( "prC2C4B: ", MappingGeneratorsImages(prC2C4B), "\n" ); 
embC4B := Embedding( C2C4B, 1 ); 
B2 := ImageElm( embC4B, B );
Print( "B2 = ", B2, "\n" ); 
embC2C := Embedding( C2C4B, 2 );
C2 := ImageElm( embC2C, C );
Print( "C2 = ", C2, "\n" ); 

## boundary map is the inclusion of C2C4B in C4AQ8 
isC2C := ImagesSource( embC2C );
imBdy := [0,0]; 
imBdy[1] := ImageElm( embC4A, ImageElm( FU, C ) ); 
imBdy[2] := ImageElm( embQ8, ImageElm( FD, B ) ); 
Bdy := GroupHomomorphismByImages( C2C4B, C4AQ8, [C2,B2], imBdy ); 
Print( "Bdy = ", MappingGeneratorsImages(Bdy), "\n" ); 

## action of C4AQ8 on C2C4B 
autC2C4B := AutomorphismGroup( C2C4B ); 
imAct := [0,0,0]; 
imA := [0,0]; 
imA[1] := ImageElm( embC2C, C^A ); 
imA[2] := B2 * ImageElm( embC2C, Comm(B,A) ); 
imAct[1] := GroupHomomorphismByImages( C2C4B, C2C4B, [C2,B2], imA ); 
imA[1] := ImageElm( embC2C, C^P ); 
imA[2] := ImageElm( embC4B, B^P ); 
imAct[2] := GroupHomomorphismByImages( C2C4B, C2C4B, [C2,B2], imA ); 
imA[1] := ImageElm( embC2C, C^Q ); 
imA[2] := ImageElm( embC4B, B^Q ); 
imAct[3] := GroupHomomorphismByImages( C2C4B, C2C4B, [C2,B2], imA ); 
if not ForAll( imAct, f -> IsBijective(f) ) then 
    Error( "imAct homs not all bijective" ); 
fi;
Act := GroupHomomorphismByImages( C4AQ8, autC2C4B, [A2,P2,Q2], imAct ); 

PCG := SemidirectProduct( C4AQ8, Act, C2C4B ); 
SetName( PCG, "PCG" ); 
prPCG := Projection( PCG ); 
Print( "prPCG: ", MappingGeneratorsImages(prPCG), "\n" ); 
embC4AQ8 := Embedding( PCG, 1 ); 
A3 := ImageElm( embC4AQ8, A2 ); 
P3 := ImageElm( embC4AQ8, P2 ); 
Q3 := ImageElm( embC4AQ8, Q2 ); 
embC2C4B := Embedding( PCG, 2 ); 
C3 := ImageElm( embC2C4B, C2 ); 
B3 := ImageElm( embC2C4B, B2 ); 
genPCG := [ C3, B3, A3, P3, Q3 ]; 
Print( "genPCG = ", genPCG, "\n" ); 

## cat1-embedding 
pce := embC4AQ8; 
## tail map 
Id := One( C4AQ8 ); 
imT := [ Id, Id, A2, P2, Q2 ]; 
T := GroupHomomorphismByImages( PCG, C4AQ8, genPCG, imT ); 
Print( "T*pce*T = T ? ", T*pce*T = T, "\n" ); 

## head map 
imH := Concatenation( imBdy, [A2,P2,Q2] ); 
Print( "imH = ", imH, "\n" );
## H := GroupHomomorphismByImages( PCG, C4AQ8, genPCG, imH ); 
H := GroupGeneralMappingByImages( PCG, C4AQ8, genPCG, imH ); 
OK := IsSingleValued( H ); 
Print( "H is single valued? ", OK, "\n" ); 
Print( "H is a group hom? ", IsGroupHomomorphism( H ), "\n" );

##  part 3 : check that isomorphisms commute with embeddings 
Print( "\nisomorphisms from a perm group to a pc-group:\n" ); 
isoq8 := GroupHomomorphismByImages( q8, Q8, [p,q], [P,Q] ); 
ok := IsBijective( isoq8 ); 
Print( "      q8 -> Q8? ", ok, "\n" ); 
isoc4aq8 := GroupHomomorphismByImages( c4aq8, C4AQ8, [a2,p2,q2], [A2,P2,Q2] ); 
ok := IsBijective( isoc4aq8 ); 
Print( "c4aq8 -> C4AQ8? ", ok, "\n" ); 
isoc2c4b := GroupHomomorphismByImages( c2c4b, C2C4B, [c2,b2], [C2,B2] ); 
ok := IsBijective( isoc2c4b ); 
Print( "c2c4b -> C2C4B? ", ok, "\n" ); 
## isoG := GroupHomomorphismByImages( G, PCG, [c3,b3,a3,p3,q3], [C3,B3,A3,P3,Q3] ); 
isoG := GroupGeneralMappingByImages( G, PCG, [c3,b3,a3,p3,q3], [C3,B3,A3,P3,Q3] ); 
ok := IsBijective( isoG ); 
Print( "      G -> PCG? ", ok, "\n" ); 
ok := IsSingleValued( isoG ); 
Print( "G -> PCG single valued? ", ok, "\n" ); 

Print( "\ncomposites of isomorphisms and embeddings:\n" ); 
ok := isoq8 * embQ8 = embq8 * isoc4aq8; 
Print( "     isoq8 * embQ8 = embq8 * isoc4aq8? ", ok, "\n" ); 
ok := prc4aq8 * isoq8 = isoc4aq8 * prC4AQ8;
Print( " prc4aq8 * isoq8 = isoc4aq8 * prC4AQ8? ", ok, "\n" );
ok := isoc4aq8 * embC4AQ8 = embc4aq8 * isoG; 
Print( "isoc4aq8 * embC4AQ8 = embc4aq8 * isoG? ", ok, "\n" );