File: perf.gd

package info (click to toggle)
gap 4r8p6-2
  • links: PTS
  • area: main
  • in suites: stretch
  • size: 33,476 kB
  • ctags: 7,663
  • sloc: ansic: 108,841; xml: 47,807; sh: 3,628; perl: 2,342; makefile: 796; asm: 62; awk: 6
file content (376 lines) | stat: -rw-r--r-- 13,047 bytes parent folder | download | duplicates (3)
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
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
#############################################################################
##
#W  perf.gd               GAP Groups Library                 Alexander Hulpke
##
##
#Y  Copyright (C)  1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,  Germany
##
##  This file contains the declarations for the Holt/Plesken library of
##  perfect groups
##


PERFRec := fail; # indicator that perf0.grp is not loaded
PERFSELECT := [];
PERFGRP := [];


#############################################################################
##
#C  IsPerfectLibraryGroup(<G>)  identifier for groups constructed from the
##                              library (used for perm->fp isomorphism)
##
##  <ManSection>
##  <Filt Name="IsPerfectLibraryGroup" Arg='G' Type='Category'/>
##
##  <Description>
##  </Description>
##  </ManSection>
##
DeclareCategory("IsPerfectLibraryGroup", IsGroup );


#############################################################################
##
#O  PerfGrpConst(<filter>,<descriptor>)
##
##  <ManSection>
##  <Oper Name="PerfGrpConst" Arg='filter,descriptor'/>
##
##  <Description>
##  </Description>
##  </ManSection>
##
DeclareConstructor("PerfGrpConst",[IsGroup,IsList]);


#############################################################################
##
#F  PerfGrpLoad(<size>)  force loading of secondary files, return index
##
##  <ManSection>
##  <Func Name="PerfGrpLoad" Arg='size'/>
##
##  <Description>
##  </Description>
##  </ManSection>
##
DeclareGlobalFunction("PerfGrpLoad");


#############################################################################
##
#A  PerfectIdentification(<G>) . . . . . . . . . . . . id. for perfect groups
##
##  <#GAPDoc Label="PerfectIdentification">
##  <ManSection>
##  <Attr Name="PerfectIdentification" Arg='G'/>
##
##  <Description>
##  This attribute is set for all groups obtained from the perfect groups
##  library and has the value <C>[<A>size</A>,<A>nr</A>]</C> if the group is obtained with
##  these parameters from the library.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute("PerfectIdentification", IsGroup );


#############################################################################
##
#F  SizesPerfectGroups()
##
##  <#GAPDoc Label="SizesPerfectGroups">
##  <ManSection>
##  <Func Name="SizesPerfectGroups" Arg=''/>
##
##  <Description>
##  This is the ordered list of all numbers up to <M>10^6</M> that occur as
##  sizes of perfect groups.
##  One can iterate over the perfect groups library with:
##  <Example><![CDATA[
##  gap> for n in SizesPerfectGroups() do
##  >      for k in [1..NrPerfectLibraryGroups(n)] do
##  >        pg := PerfectGroup(n,k);
##  >      od;
##  >    od;
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction("SizesPerfectGroups");


#############################################################################
##
#F  NumberPerfectGroups( <size> ) . . . . . . . . . . . . . . . . . . . . . .
##
##  <#GAPDoc Label="NumberPerfectGroups">
##  <ManSection>
##  <Func Name="NumberPerfectGroups" Arg='size'/>
##
##  <Description>
##  returns the number of non-isomorphic perfect groups of size <A>size</A> for
##  each positive integer  <A>size</A> up to <M>10^6</M> except for the eight  sizes
##  listed at the beginning  of  this section for  which the number is not
##  yet known. For these values as well as for any argument out of range it
##  returns <K>fail</K>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction("NumberPerfectGroups");
DeclareSynonym("NrPerfectGroups",NumberPerfectGroups);


#############################################################################
##
#F  NumberPerfectLibraryGroups( <size> )  . . . . . . . . . . . . . . . . . .
##
##  <#GAPDoc Label="NumberPerfectLibraryGroups">
##  <ManSection>
##  <Func Name="NumberPerfectLibraryGroups" Arg='size'/>
##
##  <Description>
##  returns the number of perfect groups of size <A>size</A> which are available
##  in the  library of finite perfect groups. (The purpose  of the function
##  is  to provide a simple way  to formulate a loop over all library groups
##  of a given size.)
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction("NumberPerfectLibraryGroups");
DeclareSynonym("NrPerfectLibraryGroups",NumberPerfectLibraryGroups);


#############################################################################
##
#F  PerfectGroup( [<filt>, ]<size>[, <n>] )
#F  PerfectGroup( [<filt>, ]<sizenumberpair> )
##
##  <#GAPDoc Label="PerfectGroup">
##  <ManSection>
##  <Heading>PerfectGroup</Heading>
##  <Func Name="PerfectGroup" Arg='[filt, ]size[, n]'
##   Label="for group order (and index)"/>
##  <Func Name="PerfectGroup" Arg='[filt, ]sizenumberpair'
##   Label="for a pair [ order, index ]"/>
##
##  <Description>
##  returns a group which is isomorphic to the library group specified
##  by the size number <C>[ <A>size</A>, <A>n</A> ]</C> or by the two
##  separate arguments <A>size</A> and <A>n</A>, assuming a default value of
##  <M><A>n</A> = 1</M>.
##  The optional argument <A>filt</A> defines the filter in which the group is
##  returned.
##  Possible filters so far are <Ref Func="IsPermGroup"/> and
##  <Ref Func="IsSubgroupFpGroup"/>.
##  In the latter case, the  generators and relators used coincide with those
##  given in&nbsp;<Cite Key="HP89"/>.
##  <Example><![CDATA[
##  gap> G := PerfectGroup(IsPermGroup,6048,1);
##  U3(3)
##  gap> G:=PerfectGroup(IsPermGroup,823080,2);
##  A5 2^1 19^2 C 19^1
##  gap> NrMovedPoints(G);
##  6859
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction("PerfectGroup");


#############################################################################
##
#F  DisplayInformationPerfectGroups( <size>[, <n>] )  . . . . . . . . . . . .
#F  DisplayInformationPerfectGroups( <sizenumberpair>] )  . . . . . . . . . .
##
##  <#GAPDoc Label="DisplayInformationPerfectGroups">
##  <ManSection>
##  <Heading>DisplayInformationPerfectGroups</Heading>
##  <Func Name="DisplayInformationPerfectGroups" Arg='size[, n]'
##   Label="for group order (and index)"/>
##  <Func Name="DisplayInformationPerfectGroups" Arg='sizenumberpair'
##   Label="for a pair [ order, index ]"/>
##
##  <Description>
##  <Ref Func="DisplayInformationPerfectGroups" Label="for group order (and index)"/>
##  displays some invariants of the <A>n</A>-th group of order <A>size</A>
##  from the perfect groups library.
##  <P/>
##  If no value of <A>n</A> has been specified, the invariants will be
##  displayed for all groups of size <A>size</A> available in the library.
##  <P/>
##  Alternatively, also a list of length two may be entered as the only
##  argument, with entries <A>size</A> and <A>n</A>.
##  <P/>
##  The information provided for <M>G</M> includes the following items:
##  <List>
##  <Item>
##        a headline containing the size number <C>[ <A>size</A>, <A>n</A> ]</C> of <M>G</M>
##        in the form <C><A>size</A>.<A>n</A></C> (the suffix <C>.<A>n</A></C> will be suppressed
##        if, up to isomorphism, <M>G</M> is the only perfect group of order
##        <A>size</A>),
##  </Item>
##  <Item>
##   a message if <M>G</M> is simple  or quasisimple, i.e.,
##        if the factor group of <M>G</M> by its centre is simple,
##  </Item>
##  <Item>
##   the <Q>description</Q> of  the structure of  <M>G</M> as it is
##      given by Holt and Plesken in&nbsp;<Cite Key="HP89"/> (see below),
##  </Item>
##  <Item>
##   the size of  the centre of <M>G</M>  (suppressed, if <M>G</M> is
##      simple),
##  </Item>
##  <Item>
##   the prime decomposition of the size of <M>G</M>,
##  </Item>
##  <Item>
##   orbit sizes for  a faithful permutation representation
##      of <M>G</M> which is provided by the library (see below),
##  </Item>
##  <Item>
##   a reference to each occurrence of <M>G</M> in the tables of
##      section 5.3    of  <Cite Key="HP89"/>. Each  of   these  references
##      consists of a class number and an internal number <M>(i,j)</M> under which
##      <M>G</M> is listed in that class. For some groups, there  is more than one
##      reference because these groups belong to more than one of the classes
##      in the book.
##  </Item>
##  </List>
##  <Example><![CDATA[
##  gap> DisplayInformationPerfectGroups( 30720, 3 );
##  #I Perfect group 30720:  A5 ( 2^4 E N 2^1 E 2^4 ) A
##  #I   size = 2^11*3*5  orbit size = 240
##  #I   Holt-Plesken class 1 (9,3)
##  gap> DisplayInformationPerfectGroups( 30720, 6 );
##  #I Perfect group 30720:  A5 ( 2^4 x 2^4 ) C N 2^1
##  #I   centre = 2  size = 2^11*3*5  orbit size = 384
##  #I   Holt-Plesken class 1 (9,6)
##  gap> DisplayInformationPerfectGroups( Factorial( 8 ) / 2 );
##  #I Perfect group 20160.1:  A5 x L3(2) 2^1
##  #I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 5 + 16
##  #I   Holt-Plesken class 31 (1,1) (occurs also in class 32)
##  #I Perfect group 20160.2:  A5 2^1 x L3(2)
##  #I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 7 + 24
##  #I   Holt-Plesken class 31 (1,2) (occurs also in class 32)
##  #I Perfect group 20160.3:  ( A5 x L3(2) ) 2^1
##  #I   centre = 2  size = 2^6*3^2*5*7  orbit size = 192
##  #I   Holt-Plesken class 31 (1,3)
##  #I Perfect group 20160.4:  simple group  A8
##  #I   size = 2^6*3^2*5*7  orbit size = 8
##  #I   Holt-Plesken class 26 (0,1)
##  #I Perfect group 20160.5:  simple group  L3(4)
##  #I   size = 2^6*3^2*5*7  orbit size = 21
##  #I   Holt-Plesken class 27 (0,1)
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction("DisplayInformationPerfectGroups");


#############################################################################
##
#F  SizeNumbersPerfectGroups( <factor1>, <factor2>, ... )
##
##  <#GAPDoc Label="SizeNumbersPerfectGroups">
##  <ManSection>
##  <Func Name="SizeNumbersPerfectGroups" Arg='factor1, factor2, ...'/>
##
##  <Description>
##  <Ref Func="SizeNumbersPerfectGroups"/> returns a list of pairs,
##  each entry consisting of a group order and the number of those groups in
##  the library of perfect groups that contain the specified factors
##  <A>factor1</A>, <A>factor2</A>, ...
##  among their composition factors.
##  <P/>
##  Each argument must either be the name of a simple group or an integer
##  which stands for the product of the sizes of one or more cyclic factors.
##  (In fact, the function replaces all integers among the arguments
##  by their product.)
##  <P/>
##  The following text strings are accepted as simple group names.
##  <List>
##  <Item>
##     <C>A<A>n</A></C> or <C>A(<A>n</A>)</C> for the alternating groups
##     <M>A_{<A>n</A>}</M>,
##     <M>5 \leq n \leq 9</M>, for example <C>A5</C> or <C>A(6)</C>.
##  </Item>
##  <Item>
##     <C>L<A>n</A>(<A>q</A>)</C> or <C>L(<A>n</A>,<A>q</A>)</C> for
##     PSL<M>(n,q)</M>, where
##     <M>n \in \{ 2, 3 \}</M> and <M>q</M> a prime power, ranging
##     <List>
##     <Item>
##        for <M>n = 2</M> from 4 to 125
##     </Item>
##     <Item>
##        for <M>n = 3</M> from 2 to 5
##     </Item>
##     </List>
##  </Item>
##  <Item>
##     <C>U<A>n</A>(<A>q</A>)</C> or <C>U(<A>n</A>,<A>q</A>)</C> for
##     PSU<M>(n,q)</M>, where
##     <M>n \in \{ 3, 4 \}</M> and <M>q</M> a prime power, ranging
##     <List>
##     <Item>
##        for <M>n = 3</M> from 3 to 5
##     </Item>
##     <Item>
##        for <M>n = 4</M> from 2 to 2
##     </Item>
##     </List>
##  </Item>
##  <Item>
##     <C>Sp4(4)</C> or <C>S(4,4)</C> for the symplectic group Sp<M>(4,4)</M>,
##  </Item>
##  <Item>
##     <C>Sz(8)</C> for the Suzuki group Sz<M>(8)</M>,
##  </Item>
##  <Item>
##     <C>M<A>n</A></C> or <C>M(<A>n</A>)</C> for the Mathieu groups
##     <M>M_{11}</M>, <M>M_{12}</M>, and <M>M_{22}</M>, and
##  </Item>
##  <Item>
##     <C>J<A>n</A></C> or <C>J(<A>n</A>)</C> for the Janko groups
##     <M>J_1</M> and <M>J_2</M>.
##  </Item>
##  </List>
##  <P/>
##  Note  that, for  most  of the  groups,   the  preceding list  offers  two
##  different  names in order  to  be consistent  with the  notation used  in
##  <Cite Key="HP89"/> as well as with the notation used in the
##  <Ref Func="DisplayCompositionSeries"/> command of &GAP;.
##  However, as the names are
##  compared  as text strings, you are  restricted to  the above choice. Even
##  expressions like <C>L2(2^5)</C> are not accepted.
##  <P/>
##  As the use of the term PSU<M>(n,q)</M> is not unique in the literature,
##  we mention that in this library it denotes the factor group of
##  SU<M>(n,q)</M> by its centre, where SU<M>(n,q)</M> is the group of all
##  <M>n \times n</M> unitary matrices with entries in <M>GF(q^2)</M>
##  and determinant 1.
##  <P/>
##  The purpose  of the function is  to provide a  simple way to  formulate a
##  loop over all library groups which contain certain composition factors.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction("SizeNumbersPerfectGroups");


#############################################################################
##
#E