|
#############################################################################
##
#M IsIsomorphicGroup( <G>, <H> ) . . Check if two groups G, H are isomorphic
## V0.2 3.10.94
## The return value is 'false' iff G and H are not isomorphic
##
InstallMethod(
IsIsomorphicGroup,
"test all",
true,
[IsGroup, IsGroup],
0,
function( G, H )
return not ( IsomorphismGroups( G, H ) = fail );
end );
InstallMethod(
IsIsomorphicGroup,
"both abelian",
true,
[IsGroup, IsGroup],
100,
function( G, H )
if IsAbelian(G) <> IsAbelian(H) then
return false;
else
TryNextMethod();
fi;
end );
InstallMethod(
IsIsomorphicGroup,
"same size",
true,
[IsGroup, IsGroup],
90,
function( G, H )
if Size(G) <> Size(H) then
return false;
else
TryNextMethod();
fi;
end );
InstallMethod(
IsIsomorphicGroup,
"small groups",
true,
[IsGroup, IsGroup],
85,
function ( G , H )
if Size(G)<=1000 and not (Size(G) in [256,512,768]) then
return IdGroup(G)=IdGroup(H);
else
TryNextMethod();
fi;
end );
InstallMethod(
IsIsomorphicGroup,
"centres of equal size",
true,
[IsGroup, IsGroup],
80,
function( G, H )
if Size(Centre(G)) <> Size(Centre(H)) then
return false;
else
TryNextMethod();
fi;
end );
InstallMethod(
IsIsomorphicGroup,
"commutator subgroups of equal size",
true,
[IsGroup, IsGroup],
70,
function( G, H )
if Size(DerivedSubgroup(G)) <> Size(DerivedSubgroup(H)) then
return false;
else
TryNextMethod();
fi;
end );
InstallMethod(
IsIsomorphicGroup,
"elements of the same order",
true,
[IsGroup, IsGroup],
60,
function( G, H )
if Collected( List( AsList(G), Order ) ) <>
Collected( List( AsList(H), Order ) ) then
return false;
else
TryNextMethod();
fi;
end );
#############################################################################
##
#F Subgroups ( <G> ) a list of all subgroups of <G>
Subgroups := function ( G )
local sgrps, sg;
sgrps := Flat( List( ConjugacyClassesSubgroups( G ), AsList ) );
for sg in sgrps do
SetParent( sg, Parent(G) );
od;
return sgrps;
end;
##############################################################################
##
#M OneGeneratedNormalSubgroups ( <G> )
##
## Computes a list of all normal subgroups of <G>
## that are generated by one element.
##
#InstallMethod(
# OneGeneratedNormalSubgroups,
# "default",
# true,
# [IsGroup],
# 0,
# function (G)
# return List (
# ConjugacyClasses (G),
# C -> NormalClosure (G,
# Subgroup (G, [Representative (C)])
# )
# );
# end );
#####################################################################
##
#F IsInvariantUnderMaps ( <G>, <U>, <maps>)
##
## returns true iff the subgroup <U> is invariant under
## all mappings in <maps>. All m in <maps> operate on <G>.
##
IsInvariantUnderMaps := function ( G, U, maps )
return ForAll (maps,
m -> IsSubset (U,
Images (m, U)
)
);
end;
#####################################################################
##
#M IsCharacteristic ( <G>, <U> )
## replaced by GAP-function IsCharacteristicSubgroup
##
#
#InstallMethod(
# IsCharacteristic,
# "default",
# IsIdenticalObj,
# [IsGroup, IsGroup],
# 0,
# function (G, U)
# return IsInvariantUnderMaps (G, U, Automorphisms (G));
# end );
#####################################################################
##
#M IsCharacteristicInParent ( <U> )
##
InstallMethod(
IsCharacteristicInParent,
"default",
true,
[IsGroup],
0,
function ( U )
return IsCharacteristicSubgroup( Parent(U), U );
end );
#####################################################################
##
#M IsFullinvariant ( <G>, <U> )
##
InstallMethod(
IsFullinvariant,
"default",
IsIdenticalObj,
[IsGroup, IsGroup],
0,
function (G, U)
return IsInvariantUnderMaps (G, U, Endomorphisms (G));
end );
#####################################################################
##
#M IsFullinvariantInParent ( <U> )
##
InstallMethod(
IsFullinvariantInParent,
"default",
true,
[IsGroup],
0,
function ( U )
return IsFullinvariant( Parent(U), U );
end );
#####################################################################
##
#M AsPermGroup
InstallMethod(
AsPermGroup,
"perm groups",
true,
[IsPermGroup],
1000,
G -> G );
InstallMethod( AsPermGroup, [IsGroup],
G -> Image( IsomorphismPermGroup( G ) ) );
#####################################################################
##
#M PrintTable2
InstallMethod(
PrintTable2,
"groups",
true,
[IsGroup, IsString],
0,
function ( G, mode )
# mode is simply ignored here!
local elms, # the elements of the group
n, # the size of the group
symbols, # a list of the symbols for the elements of the group
tw, # the width of a table
spc, # local function which prints the right number of spaces
bar, # local function for printing the right length of the bar
ind, # help variable, an index
i,j, # loop variables
max; # length of the longest symbol string
elms := AsSSortedList( G );
n := Length( elms );
symbols := Symbols( G );
max := Maximum( List( symbols, Length ) );
# compute the number of characters per line required for the table
tw := (max+1)*(n+1) + 2;
if SizeScreen()[1] - 3 < tw then
Print( "The table of a group of order ", n, " will not ",
"look\ngood on a screen with ", SizeScreen()[1], " characters per ",
"line.\nHowever, you may want to set your line length to a ",
"greater\nvalue by using the GAP function 'SizeScreen'.\n" );
return;
fi;
spc := function( i )
return String(
Concatenation( List( [Length(symbols[i])..max+1], j -> " " ) )
);
end;
bar := function()
return String( Concatenation( List( [0..max+1], i -> "-" ) ) );
end;
Print( "Let:\n" );
for i in [1..n] do Print( symbols[i], " := ", elms[i], "\n" ); od;
# print the addition table
Print("\n");
for i in [1..max+1] do Print(" "); od;
Print( "* | " );
for i in [1..n] do Print( symbols[i], spc(i) ); od;
Print( "\n ", bar() ); for i in [1..n] do Print( bar() ); od;
for i in [1..n] do
Print( "\n ", symbols[i], spc(i), "| " );
for j in [1..n] do
ind := Position( elms, elms[i] * elms[j] );
Print( symbols[ ind ], spc(ind) );
od;
od;
Print("\n");
end );
#####################################################################
##
#F PrintTable( <domain> [, <mode>] ) print the operation table of
## a group or a nearring.
## With nearrings <mode> can be used to specify which
## information should be printed:
## 'e': information about the elements
## 'a': addition table
## 'm': multiplication table
## the default value for mode is "eam"
InstallGlobalFunction(
PrintTable,
function ( arg )
if Length(arg)=1 then
PrintTable2(arg[1],"eam");
else
PrintTable2(arg[1],arg[2]);
fi;
end);
#####################################################################
##
#F IdTWGroup ( <G> ) returns the Thomas/Wood group classification
## number of the group <G> as a pair of integers
IdTWGroup := function ( G )
local table, id;
if Size(G)>32 then
Error("group must be of order at most 32");
fi;
# table is a translation table between GAP and Thomas/Wood notation
table := [
[1],
[1],
[1],
[1,2],
[1],
[2,1],
[1],
[1,2,4,5,3],
[1,2],
[2,1],
[1],
[5,1,4,3,2],
[1],
[2,1],
[1],
[1,3,9,10,2,11,12,13,14,4,6,7,8,5],
[1],
[4,1,3,5,2],
[1],
[4,1,5,3,2],
[2,1],
[2,1],
[1],
[14,1,13,11,9,10,6,15,2,7,8,12,5,4,3],
[1,2],
[2,1],
[1,2,4,5,3],
[4,1,3,2],
[1],
[3,2,4,1],
[1],
[1,18,3,19,20,46,47,48,27,28,31,21,30,29,32,2,22,49,50,51,5,11,
12,16,14,15,33,36,37,38,39,40,41,34,35,4,13,17,23,24,25,26,44,
45,6,8,9,10,42,43,7],
];
# ask GAP
id := IdGroup(G);
# translate
return [id[1],table[id[1]][id[2]]];
end;
#####################################################################
##
#M RepresentativesModNormalSubgroup
##
InstallMethod(
RepresentativesModNormalSubgroup,
"default",
IsIdenticalObj,
[IsGroup, IsGroup],
0,
function ( G, N )
return List( RightCosets( G, N ), Representative );
end );
#####################################################################
##
#M NontrivialRepresentativesModNormalSubgroup
##
InstallMethod(
NontrivialRepresentativesModNormalSubgroup,
"default",
IsIdenticalObj,
[IsGroup, IsGroup],
0,
function ( G, N )
local RmNSg;
RmNSg := RepresentativesModNormalSubgroup( G, N );
return RmNSg{[2..Length(RmNSg)]};
end );
#############################################################################
##
#M ScottLength
##
#PrincipalSeries := function (G)
#
# returns one principal series of G
#
# local N, i, principalSeries, lastadded;
#
# N := NormalSubgroups (G);
# Print ("\n1\n");
# principalSeries := [G];
# lastadded := G;
#
# i := Length (N) - 1;
# while i >= 1 do
# if IsSubgroup (lastadded, N[i]) then
# Add (principalSeries, N[i]);
# lastadded := N[i];
# fi;
# i := i - 1;
# od;
#
# return principalSeries;
#end;
InstallGlobalFunction(
ScottSigma,
function (G, N1, N2)
if (not IsSubgroup (N1, N2)) or (Size (N2) = Size (N1)) then
Error ("length.x : Wrong order of parameters");
fi;
if IsSubgroup (N2, CommutatorSubgroup (G, N1)) then
return Size (N1) / Size (N2);
else
return 1;
fi;
end );
InstallMethod(
ScottLength,
"abelian",
true,
[IsGroup and IsAbelian],
0,
function ( G )
return Exponent( G );
end );
BindGlobal( "FindKInPGroup",
function ( PG )
# find a k in a p-group PG s.t. kx = [x,pi] for a pi and k
local info, # the return-value
p, # which p-group
ExpG, # the exponent of PG is p^ExpG
ExpZ, # the same for Z(PG)
ExpGz, # the same for PG/Z(PG)
ExpGk, # the same for PG/K(PG)
LowerBound,
UpperBound, # bounds for k
possibilities,# possible values for k
k, # a possible value for k
gens, l, indices, indx, pi, orders;
info := rec (exponent := Exponent(PG));
if IsAbelian(PG)
or IsPrime(Exponent(PG)) # Corollary 4
or Exponent(PG)=4 # Corollary 8
then
info.IsSmall := false;
info.k := Exponent(PG);
return info;
fi;
p := Factors(Exponent(PG))[1];
ExpG := Length(Factors(Exponent(PG)));
UpperBound := ExpG - 1;
LowerBound := 1;
# Corollary 7
if p=2 then
LowerBound := Maximum(LowerBound,EuclideanQuotient(ExpG+2,2));
else
LowerBound := Maximum(LowerBound,EuclideanQuotient(ExpG+1,2));
fi;
# Corollary 11 (replaces C. 6 and C. 10) & 12
ExpGz := Length(Factors(Exponent(PG/Centre(PG))));
LowerBound := Maximum(LowerBound,ExpG-ExpGz);
if LowerBound > UpperBound then
info.IsSmall := false;
info.k := p^ExpG;
return info;
fi;
# Proposition 7
ExpGk := Length(Factors(Exponent(PG/DerivedSubgroup(PG))));
LowerBound := Maximum(LowerBound,ExpGk);
if LowerBound > UpperBound then
info.IsSmall := false;
info.k := p^ExpG;
return info;
fi;
# Trying Out the k's beginning with the smallest
possibilities := List([LowerBound..UpperBound],i->p^i);
for k in possibilities do
# Proposition 9
if IsSubset(Centre(PG), Set(List(PG,x->x^k)))
and
# Proposition 14
ForAll(Combinations(Filtered(GeneratorsOfGroup(PG),
gen->gen^k<>Identity(PG)),2),
gen->Comm(gen[1],gen[2])^(k*(k-1)/2)=Identity(PG))
then
gens := GeneratorsOfGroup(PG);
l := Length(gens);
orders := List( gens, Order );
indices := Cartesian(List(orders, o -> [1..o]));
for indx in indices do
pi := Product(List([1..l], i -> gens[i]^indx[i]));
if pi^k=Identity(PG) and
pi^(p^(ExpG-1)/k)<>Identity(PG) and
ForAll(GeneratorsOfGroup(PG),gen->gen^k=Comm(gen,pi)) then
info.IsSmall := true;
info.k := k;
return info;
fi;
od;
fi;
od;
info.IsSmall := false;
info.k := Exponent(PG);
return info;
end );
BindGlobal( "FindKInGroup",
function ( G )
# find a possible k in the group G
local pees, # values for p with nontrivial p-Sylow subgroup
SylowSgps, # the corresponding p-Sylow subgroups
SSgInfo; # information about possible k's
# for the Sylow subgroups
pees := Set(Factors(Size(G)));
SylowSgps := List(pees,p->SylowSubgroup(G,p));
SSgInfo := List(SylowSgps,ssg->FindKInPGroup(ssg));
return Product(List(SSgInfo,info->info.k));
end );
InstallMethod(
ScottLength,
"nilpotent of class 2",
true,
[IsGroup],
2,
function ( G )
if not IsNilpotent( G ) or NilpotencyClassOfGroup( G ) <> 2 then
TryNextMethod();
fi;
return FindKInGroup( G );
end );
InstallMethod(
ScottLength,
"default",
true,
[IsGroup],
0,
function ( G )
# Computes the length of a group according to
# Scott, Monatshefte f. Math 73, 250 - 267 (1969)
#
local primes, principalSeries, product, i, N,
n, p, IG, length, primepowerList;
# N := NormalSubgroups (G);
# principalSeries := PrincipalSeries (G);
principalSeries := ChiefSeries (G);
product := 1;
for i in [1..Length (principalSeries) - 1] do
product := product *
ScottSigma( G, principalSeries[i], principalSeries[i + 1] );
od;
primes := Set( FactorsInt(product) );
if product <> 1 then
IG := InnerAutomorphismNearRing( G );
length := 1;
for p in primes do
n := 1;
while
MOD( Exponent(G), p^(n+1) ) = 0 and
MOD( product, p ^ (n+1) ) = 0 and
( Size( InnerAutomorphismNearRing(
DirectProduct( G, CyclicGroup( p^(n+1) ) )
) )
=
Size (IG)
) do
n := n+1;
od;
# now n has the maximal possible value
length := length * p^n;
od;
return length;
else
return 1;
fi;
end );
#############################################################################
##
#M TWGroup
##
InstallMethod(
TWGroup,
"default",
true,
[IsInt and IsPosRat, IsInt and IsPosRat],
0,
function ( size, nr )
if size = 1 then
if nr = 1 then
return Group( () );
fi;
elif size = 2 then
if nr = 1 then
return GTW2_1;
fi;
elif size = 3 then
if nr = 1 then
return GTW3_1;
fi;
elif size = 4 then
if nr = 1 then
return GTW4_1;
elif nr = 2 then
return GTW4_2;
fi;
elif size = 5 then
if nr = 1 then
return GTW5_1;
fi;
elif size = 6 then
if nr = 1 then
return GTW6_1;
elif nr = 2 then
return GTW6_2;
fi;
elif size = 7 then
if nr = 1 then
return GTW7_1;
fi;
elif size = 8 then
if nr = 1 then
return GTW8_1;
elif nr = 2 then
return GTW8_2;
elif nr = 3 then
return GTW8_3;
elif nr = 4 then
return GTW8_4;
elif nr = 5 then
return GTW8_5;
fi;
elif size = 9 then
if nr = 1 then
return GTW9_1;
elif nr = 2 then
return GTW9_2;
fi;
elif size = 10 then
if nr = 1 then
return GTW10_1;
elif nr = 2 then
return GTW10_2;
fi;
elif size = 11 then
if nr = 1 then
return GTW11_1;
fi;
elif size = 12 then
if nr = 1 then
return GTW12_1;
elif nr = 2 then
return GTW12_2;
elif nr = 3 then
return GTW12_3;
elif nr = 4 then
return GTW12_4;
elif nr = 5 then
return GTW12_5;
fi;
elif size = 13 then
if nr = 1 then
return GTW13_1;
fi;
elif size = 14 then
if nr = 1 then
return GTW14_1;
elif nr = 2 then
return GTW14_2;
fi;
elif size = 15 then
if nr = 1 then
return GTW15_1;
fi;
elif size = 16 then
if nr = 1 then
return GTW16_1;
elif nr = 2 then
return GTW16_2;
elif nr = 3 then
return GTW16_3;
elif nr = 4 then
return GTW16_4;
elif nr = 5 then
return GTW16_5;
elif nr = 6 then
return GTW16_6;
elif nr = 7 then
return GTW16_7;
elif nr = 8 then
return GTW16_8;
elif nr = 9 then
return GTW16_9;
elif nr = 10 then
return GTW16_10;
elif nr = 11 then
return GTW16_11;
elif nr = 12 then
return GTW16_12;
elif nr = 13 then
return GTW16_13;
elif nr = 14 then
return GTW16_14;
fi;
elif size = 17 then
if nr = 1 then
return GTW17_1;
fi;
elif size = 18 then
if nr = 1 then
return GTW18_1;
elif nr = 2 then
return GTW18_2;
elif nr = 3 then
return GTW18_3;
elif nr = 4 then
return GTW18_4;
elif nr = 5 then
return GTW18_5;
fi;
elif size = 19 then
if nr = 1 then
return GTW19_1;
fi;
elif size = 20 then
if nr = 1 then
return GTW20_1;
elif nr = 2 then
return GTW20_2;
elif nr = 3 then
return GTW20_3;
elif nr = 4 then
return GTW20_4;
elif nr = 5 then
return GTW20_5;
fi;
elif size = 21 then
if nr = 1 then
return GTW21_1;
elif nr = 2 then
return GTW21_2;
fi;
elif size = 22 then
if nr = 1 then
return GTW22_1;
elif nr = 2 then
return GTW22_2;
fi;
elif size = 23 then
if nr = 1 then
return GTW23_1;
fi;
elif size = 24 then
if nr = 1 then
return GTW24_1;
elif nr = 2 then
return GTW24_2;
elif nr = 3 then
return GTW24_3;
elif nr = 4 then
return GTW24_4;
elif nr = 5 then
return GTW24_5;
elif nr = 6 then
return GTW24_6;
elif nr = 7 then
return GTW24_7;
elif nr = 8 then
return GTW24_8;
elif nr = 9 then
return GTW24_9;
elif nr = 10 then
return GTW24_10;
elif nr = 11 then
return GTW24_11;
elif nr = 12 then
return GTW24_12;
elif nr = 13 then
return GTW24_13;
elif nr = 14 then
return GTW24_14;
elif nr = 15 then
return GTW24_15;
fi;
elif size = 25 then
if nr = 1 then
return GTW25_1;
elif nr = 2 then
return GTW25_2;
fi;
elif size = 26 then
if nr = 1 then
return GTW26_1;
elif nr = 2 then
return GTW26_2;
fi;
elif size = 27 then
if nr = 1 then
return GTW27_1;
elif nr = 2 then
return GTW27_2;
elif nr = 3 then
return GTW27_3;
elif nr = 4 then
return GTW27_4;
elif nr = 5 then
return GTW27_5;
fi;
elif size = 28 then
if nr = 1 then
return GTW28_1;
elif nr = 2 then
return GTW28_2;
elif nr = 3 then
return GTW28_3;
elif nr = 4 then
return GTW28_4;
fi;
elif size = 29 then
if nr = 1 then
return GTW29_1;
fi;
elif size = 30 then
if nr = 1 then
return GTW30_1;
elif nr = 2 then
return GTW30_2;
elif nr = 3 then
return GTW30_3;
elif nr = 4 then
return GTW30_4;
fi;
elif size = 31 then
if nr = 1 then
return GTW31_1;
fi;
elif size = 32 then
if nr = 1 then
return GTW32_1;
elif nr = 2 then
return GTW32_2;
elif nr = 3 then
return GTW32_3;
elif nr = 4 then
return GTW32_4;
elif nr = 5 then
return GTW32_5;
elif nr = 6 then
return GTW32_6;
elif nr = 7 then
return GTW32_7;
elif nr = 8 then
return GTW32_8;
elif nr = 9 then
return GTW32_9;
elif nr = 10 then
return GTW32_10;
elif nr = 11 then
return GTW32_11;
elif nr = 12 then
return GTW32_12;
elif nr = 13 then
return GTW32_13;
elif nr = 14 then
return GTW32_14;
elif nr = 15 then
return GTW32_15;
elif nr = 16 then
return GTW32_16;
elif nr = 17 then
return GTW32_17;
elif nr = 18 then
return GTW32_18;
elif nr = 19 then
return GTW32_19;
elif nr = 20 then
return GTW32_20;
elif nr = 21 then
return GTW32_21;
elif nr = 22 then
return GTW32_22;
elif nr = 23 then
return GTW32_23;
elif nr = 24 then
return GTW32_24;
elif nr = 25 then
return GTW32_25;
elif nr = 26 then
return GTW32_26;
elif nr = 27 then
return GTW32_27;
elif nr = 28 then
return GTW32_28;
elif nr = 29 then
return GTW32_29;
elif nr = 30 then
return GTW32_30;
elif nr = 31 then
return GTW32_31;
elif nr = 32 then
return GTW32_32;
elif nr = 33 then
return GTW32_33;
elif nr = 34 then
return GTW32_34;
elif nr = 35 then
return GTW32_35;
elif nr = 36 then
return GTW32_36;
elif nr = 37 then
return GTW32_37;
elif nr = 38 then
return GTW32_38;
elif nr = 39 then
return GTW32_39;
elif nr = 40 then
return GTW32_40;
elif nr = 41 then
return GTW32_41;
elif nr = 42 then
return GTW32_42;
elif nr = 43 then
return GTW32_43;
elif nr = 44 then
return GTW32_44;
elif nr = 45 then
return GTW32_45;
elif nr = 46 then
return GTW32_46;
elif nr = 47 then
return GTW32_47;
elif nr = 48 then
return GTW32_48;
elif nr = 49 then
return GTW32_49;
elif nr = 50 then
return GTW32_50;
elif nr = 51 then
return GTW32_51;
fi;
fi;
Error( "sorry, no group ",size,"/",nr );
end );
#####################################################################
##
#M Symbols
##
InstallMethod(
Symbols,
"groups",
true,
[IsGroup],
0,
function ( G )
return List( [1..Size(G)], i ->
String( Concatenation( "g", String(i-1) ) ) );
end );
#############################################################################
##
#M FastImageOfProjection
InstallMethod(
FastImageOfProjection,
"perm groups - very dangerous !!!",
true,
[IsGroup and IsPermGroup, IsMultiplicativeElementWithInverse,
IsInt and IsPosRat],
0,
function( DP, dpElm, i )
local G, d, list, inv_perm;
G := DirectProductInfo( DP ).groups[1];
inv_perm := (DirectProductInfo( DP ).perms[1])^-1;
d := Length( MovedPoints( G ) );
# if dpElm = () then
# list := [];
# else
list := ListPerm( dpElm );
# fi;
if Length(list) < d*Size(G) then
list := Concatenation( list, [Length(list)+1..d*Size(G)] );
fi;
list := List( list{[d*(i-1)+1..d*i]}, k -> k-d*(i-1) );
return PermList( list )^inv_perm;
end );
InstallMethod(
FastImageOfProjection,
"standard",
true,
[IsGroup, IsMultiplicativeElementWithInverse, IsInt and IsPosRat],
0,
function( DP, dpElm, i )
return dpElm^Projection( DP, i );
end );
[ Dauer der Verarbeitung: 0.41 Sekunden
(vorverarbeitet)
]
|
