1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
|
#############################################################################
##
#W initmat.gi AutPGrp package Bettina Eick
##
#H @(#)$Id: initmat.gi,v 1.2 2002/11/19 13:36:38 gap Exp $
##
Revision.("autpgrp/gap/initmat_gi") :=
"@(#)$Id: initmat.gi,v 1.2 2002/11/19 13:36:38 gap Exp $";
#############################################################################
##
#F TwoStepCentralizersByLcs( G )
##
## The two-step-centralizers of lower p-central series of G.
##
TwoStepCentralizersByLcs := function( G )
local pcgs, first, p, field, list, i, f, m, n, max, pcgsN, pcgsM,
pcgsH, gensL, gensC, pcgsR, new;
# set up
pcgs := SpecialPcgs( G );
first := LGFirst( pcgs );
p := PrimePGroup( G );
field := GF(p);
list := [];
max := Length(pcgs);
# run through lower p-central series
for i in [3..Length(first)] do
f := first[i-2];
m := first[i-1];
n := first[i];
pcgsN := InducedPcgsByPcSequenceNC( pcgs, pcgs{[f..max]} );
pcgsM := InducedPcgsByPcSequenceNC( pcgs, pcgs{[m..max]} );
pcgsH := InducedPcgsByPcSequenceNC( pcgs, pcgs{[n..max]} );
gensL := pcgsN mod pcgsM;
gensC := pcgs mod pcgsM;
pcgsR := pcgsM mod pcgsH;
new := NextStepCentralizer( gensL, gensC, pcgsR, field );
Append(new, pcgsM );
#new := InducedPcgsByPcSequenceNC( pcgs, new );
new := InducedPcgsByGeneratorsNC( pcgs, new );
Add( list, SubgroupByPcgs( G, new ) );
od;
return list;
end;
#############################################################################
##
#F OmegaSubgroupsByLcs( G )
##
## The preimages of Omega-subgroups of G_i for all factors G_i of the lower
## p-central series of G.
##
OmegaSubgroupsByLcs := function( G )
local pcgs, first, p, field, list, max, i, pcgsN, N, hom, F, ser, specF;
# catch the trivial case
p := PrimePGroup( G );
pcgs := SpecialPcgs( G );
if ForAll( pcgs, x -> Order(x) = p ) then return []; fi;
# set up
first := LGFirst( pcgs );
field := GF(p);
list := [];
max := Length(pcgs);
# run through lower p-central series
for i in [2..Length(first)] do
pcgsN := InducedPcgsByPcSequenceNC( pcgs, pcgs{[first[i]..max]} );
N := SubgroupByPcgs( G, pcgsN );
hom := NaturalHomomorphismByNormalSubgroupNC( G, N );
F := Image( hom );
specF := SpecialPcgs(F);
if ForAny( specF, x -> Order(x) > p ) and Size(F) < 10000 then
ser := OmegaSeries( F );
ser := List( ser, x -> PreImage( hom, x ) );
Append( list, ser );
fi;
od;
return list;
end;
#############################################################################
##
#F PGCharSubgroups( G )
##
PGCharSubgroups := function(G)
local cent, omega;
cent := TwoStepCentralizersByLcs( G );
omega := OmegaSubgroupsByLcs( G );
return Union( cent, omega );
end;
#############################################################################
##
#F FrattiniQuotientBase( <spec>, <U> )
##
FrattiniQuotientBase := function( spec, U )
local r, frat, pcgs, subU, base;
r := LGFirst(spec)[2];
frat := InducedPcgsByPcSequenceNC( spec, spec{[r..Length(spec)]} );
pcgs := spec mod frat;
subU := Filtered(InducedPcgs(spec, U), x -> DepthOfPcElement(spec,x)<r);
base := List( subU, x -> ExponentsOfPcElement( pcgs, x ) );
return base;
end;
#############################################################################
##
#F InitAutomorphismGroupChar( G )
##
InitAutomorphismGroupChar := function( G )
local r, p, chars, bases, S, H, A, z, spec, kern;
Info( InfoAutGrp, 2, " init automorphism group : Char ");
# set up
r := RankPGroup( G );
p := PrimePGroup( G );
z := One(GF(p));
spec := SpecialPcgs( G );
# compute characteristic subgroups
Info( InfoAutGrp, 3, " compute characteristic subgroups ");
chars := PGCharSubgroups( G );
bases := List( chars, x -> FrattiniQuotientBase( spec, x ) ) * z;
# compute the matrixgroup stabilising all subspaces in chain
Info( InfoAutGrp, 3, " compute stabilizer ");
S := StabilizingMatrixGroup( bases, r, p );
# the Frattini Quotient
H := FrattiniQuotientPGroup( G );
kern := InitAgAutos( H, p );
# the aut group
A := rec( );
A.glAutos := InitGlAutos( H, GeneratorsOfGroup(S) );
A.glOrder := Size(S) / Product( kern.rels );
A.glOper := GeneratorsOfGroup(S);
Assert(1,IsInt(A.glOrder));
A.agAutos := kern.auts;
A.agOrder := kern.rels;
A.one := IdentityPGAutomorphism( H );
A.group := H;
A.size := A.glOrder * Product( A.agOrder );
# try to construct perm rep
NiceInitGroup( A, true );
return A;
end;
#############################################################################
##
#F InitAutomorphismGroupFull( G )
##
InitAutomorphismGroupFull := function( G )
local r, p, S, H, A, kern;
Info( InfoAutGrp, 2, " init automorphism group : Full ");
# set up
r := RankPGroup( G );
p := PrimePGroup( G );
S := GL(r, p);
H := FrattiniQuotientPGroup( G );
kern := InitAgAutos( H, p );
# the aut group
A := rec( );
A.glAutos := InitGlAutos( H, GeneratorsOfGroup(S) );
A.glOrder := Size(S) / Product( kern.rels );
A.glOper := GeneratorsOfGroup( S );
Assert(1,IsInt(A.glOrder));
A.agAutos := kern.auts;
A.agOrder := kern.rels;
A.one := IdentityPGAutomorphism( H );
A.group := H;
A.size := A.glOrder * Product( A.agOrder );
# try to compute perm rep
NiceInitGroup( A, false );
return A;
end;
|
