File: CHAP002.htm

package info (click to toggle)
gap-fga 1.4.0-2
  • links: PTS
  • area: main
  • in suites: bookworm, bullseye
  • size: 588 kB
  • sloc: makefile: 106; sh: 12
file content (322 lines) | stat: -rw-r--r-- 14,608 bytes parent folder | download
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
<html><head><title>[FGA] 2 Functionality of the FGA package</title></head>
<body text="#000000" bgcolor="#ffffff">
[<a href = "chapters.htm">Up</a>] [<a href ="CHAP001.htm">Previous</a>] [<a href ="CHAP003.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
<h1>2 Functionality of the FGA package</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP002.htm#SECT001">New operations for free groups</a>
<li> <A HREF="CHAP002.htm#SECT002">Method installations</a>
<li> <A HREF="CHAP002.htm#SECT003">Constructive membership test</a>
<li> <A HREF="CHAP002.htm#SECT004">Automorphism groups of free groups</a>
</ol><p>
<p>
<a name = "I0"></a>

This chapter describes methods available from the <font face="Gill Sans,Helvetica,Arial">FGA</font> package.
<p>
In the following, let <var>f</var> be a free group created by <code>FreeGroup(</code><var>n</var><code>)</code>,
and let <var>u</var>, <var>u1</var> and <var>u2</var> be finitely generated subgroups of <var>f</var>
created by <code>Group</code> or <code>Subgroup</code>, or computed from some other subgroup
of <var>f</var>.  Let <var>elm</var> be an element of <var>f</var>.
<p>
For example:
<p>
<pre>
gap&gt; f := FreeGroup( 2 );                                             
&lt;free group on the generators [ f1, f2 ]&gt;
gap&gt; u := Group( f.1^2, f.2^2, f.1*f.2 );
Group([ f1^2, f2^2, f1*f2 ])
gap&gt; u1 := Subgroup( u, [f.1^2, f.1^4*f.2^6] );
Group([ f1^2, f1^4*f2^6 ])
gap&gt; elm := f.1;
f1
gap&gt; u2 := Normalizer( u, elm );
Group([ f1^2 ])
</pre>
<p>
<p>
<h2><a name="SECT001">2.1 New operations for free groups</a></h2>
<p><p>
These new operations are available for finitely generated subgroups of
free groups:
<p>
<a name = "SSEC001.1"></a>
<li><code>FreeGeneratorsOfGroup( </code><var>u</var><code> ) A</code>
<p>
returns a list of free generators of the finitely generated subgroup
<var>u</var> of a free group.
<p>
The elements in this list form an N-reduced set.  In addition to
being a free (and thus minimal) generating set for <var>u</var>, this means
that whenever <var>v1</var>, <var>v2</var> and <var>v3</var> are elements or inverses of elements
of this list, then
<p>
<ul>
  <li>
    <i>v</i>1 <i>v</i>2  &#8800; 1 implies &#124;<i>v</i>1 <i>v</i>2 &#124; &#8805; max(&#124;<i>v</i>1 &#124;, &#124;<i>v</i>2 &#124;), and
  <li>
    <i>v</i>1 <i>v</i>2  &#8800; 1 and <i>v</i>2 <i>v</i>3  &#8800; 1 implies
    &#124;<i>v</i>1 <i>v</i>2 <i>v</i>3 &#124;  &gt;  &#124;<i>v</i>1 &#124; &#8722; &#124;<i>v</i>2 &#124; + &#124;<i>v</i>3 &#124;
</ul>
<p>
hold, where &#124;&#183;&#124; denotes the word length.
<p>
<a name = "SSEC001.2"></a>
<li><code>RankOfFreeGroup( </code><var>u</var><code> ) A</code>
<a name = "SSEC001.2"></a>
<li><code>Rank( </code><var>u</var><code> ) O</code>
<p>
returns the rank of the finitely generated subgroup <var>u</var> of a free
group.
<p>
<a name = "SSEC001.3"></a>
<li><code>CyclicallyReducedWord( </code><var>elm</var><code> ) O</code>
<p>
returns the cyclically reduced form of <var>elm</var>.
<p>
<p>
<h2><a name="SECT002">2.2 Method installations</a></h2>
<p><p>
This section lists operations that are already known to <font face="Gill Sans,Helvetica,Arial">GAP</font>.
<font face="Gill Sans,Helvetica,Arial">FGA</font> installs new methods for them so that they can also be used
with free groups.
In cases where <font face="Gill Sans,Helvetica,Arial">FGA</font> installs methods that are usually only used
internally, user functions are shown instead.
<p>
<a name = "SSEC002.1"></a>
<li><code>Normalizer( </code><var>u1</var><code>, </code><var>u2</var><code> ) O</code>
<li><code>Normalizer( </code><var>u</var><code>, </code><var>elm</var><code> ) O</code>
<p>
The first variant returns the normalizer of the finitely generated
subgroup <var>u2</var> in <var>u1</var>.
<p>
The second variant returns the normalizer of &#9001;<i>elm</i>  &#9002;
in the finitely generated subgroup <var>u</var> (see <a href="../../../doc/ref/chap39.html#X804F0F037F06E25E">Normalizer</a> in the
Reference Manual).
<p>
<a name = "SSEC002.2"></a>
<li><code>RepresentativeAction( </code><var>u</var><code>, </code><var>d</var><code>, </code><var>e</var><code> ) O</code>
<a name = "SSEC002.2"></a>
<li><code>IsConjugate( </code><var>u</var><code>, </code><var>d</var><code>, </code><var>e</var><code> ) O</code>
<p>
<code>RepresentativeAction</code> returns an element  <i>r</i>   &#8712; <i>u</i>  ,
where <var>u</var> is a finitely generated subgroup of a free group, such
that <i>d</i> <sup><i>r</i> </sup>=<i>e</i> , or fail, if no such <var>r</var> exists.  <var>d</var> and <var>e</var> may
be elements or subgroups of <var>u</var>.
<p>
<code>IsConjugate</code> returns a boolean indicating whether such an element <var>r</var>
exists.
<p>
<a name = "SSEC002.3"></a>
<li><code>Centralizer( </code><var>u</var><code>, </code><var>u2</var><code> ) O</code>
<li><code>Centralizer( </code><var>u</var><code>, </code><var>elm</var><code> ) O</code>
<p>
returns the centralizer of <var>u2</var> or <var>elm</var> in the finitely generated
subgroup <var>u</var> of a free group.
<p>
<a name = "SSEC002.4"></a>
<li><code>Index( </code><var>u1</var><code>, </code><var>u2</var><code> ) O</code>
<a name = "SSEC002.4"></a>
<li><code>IndexNC( </code><var>u1</var><code>, </code><var>u2</var><code> ) O</code>
<p>
return the index of <var>u2</var> in <var>u1</var>, where <var>u1</var> and <var>u2</var> are finitely
generated subgroups of a free group.  The first variant returns
fail if <var>u2</var> is not a subgroup of <var>u1</var>, the second may return
anything in this case.
<p>
<a name = "SSEC002.5"></a>
<li><code>Intersection( </code><var>u1</var><code>, </code><var>u2</var><code> ...) F</code>
<p>
returns the intersection of <var>u1</var> and <var>u2</var>, where <var>u1</var> and <var>u2</var> are
finitely generated subgroups of a free group.
<p>
<a name = "SSEC002.6"></a>
<li><code></code><var>elm</var><code> in </code><var>u</var><code> O</code>
<p>
tests whether <var>elm</var> is contained in the finitely generated subgroup
<var>u</var> of a free group.
<p>
<a name = "SSEC002.7"></a>
<li><code>IsSubgroup( </code><var>u1</var><code>, </code><var>u2</var><code> ) F</code>
<p>
tests whether <var>u2</var> is a subgroup of <var>u1</var>, where <var>u1</var> and <var>u2</var> are finitely
generated subgroups of a free group.
<p>
<a name = "SSEC002.8"></a>
<li><code></code><var>u1</var><code> = </code><var>u2</var><code> O</code>
<p>
test whether the two finitely generated subgroups <var>u1</var> and <var>u2</var> of a
free group are equal.
<p>
<a name = "SSEC002.9"></a>
<li><code>MinimalGeneratingSet( </code><var>u</var><code> ) A</code>
<a name = "SSEC002.9"></a>
<li><code>SmallGeneratingSet( </code><var>u</var><code> ) A</code>
<a name = "SSEC002.9"></a>
<li><code>GeneratorsOfGroup( </code><var>u</var><code> ) A</code>
<p>
return generating sets for the finitely generated subgroup <var>u</var> of a
free group.  <code>MinimalGeneratingSet</code> and <code>SmallGeneratingSet</code> return
the same free generators as <code>FreeGeneratorsOfGroup</code>, which are in
fact a minimal generating set.  <code>GeneratorsOfGroup</code> also returns these
generators, if no other generators were stored at creation time.
<p>
<p>
<h2><a name="SECT003">2.3 Constructive membership test</a></h2>
<p><p>
It is not only possible to test whether an element is in a finitely
generated subgroup of free group, this can also be done
constructively.  The idiomatic way to do so is by using a
homomorphism.
<p>
Here is an example that computes how to write <code>f.1^2</code> in the
generators <code>a=f1^2*f2^2</code> and <code>b=f.1^2*f.2</code>, checks the result, and
then tries to write <code>f.1</code> in the same generators:
<p>
<pre>
gap&gt; f := FreeGroup( 2 );
&lt;free group on the generators [ f1, f2 ]&gt;
gap&gt; ua := f.1^2*f.2^2;; ub := f.1^2*f.2;;
gap&gt; u := Group( ua, ub );;
gap&gt; g := FreeGroup( "a", "b" );;
gap&gt; hom := GroupHomomorphismByImages( g, u,
&gt;             GeneratorsOfGroup(g),
&gt;             GeneratorsOfGroup(u) );
[ a, b ] -&gt; [ f1^2*f2^2, f1^2*f2 ]
gap&gt; # how can f.1^2 be expressed?
gap&gt; PreImagesRepresentative( hom, f.1^2 );
b*a^-1*b
gap&gt; last ^ hom; # check this
f1^2
gap&gt; ub * ua^-1 * ub; # another check
f1^2
gap&gt; PreImagesRepresentative( hom, f.1 ); # try f.1
fail
gap&gt; f.1 in u;
false
</pre>
<p>
There are also lower level operations to get the same results.
<p>
<a name = "SSEC003.1"></a>
<li><code>AsWordLetterRepInGenerators( </code><var>elm</var><code>, </code><var>u</var><code> ) O</code>
<a name = "SSEC003.1"></a>
<li><code>AsWordLetterRepInFreeGenerators( </code><var>elm</var><code>, </code><var>u</var><code> ) O</code>
<p>
return a letter representation
(see Section&nbsp;<a href="../../../doc/ref/chap37.html#X80A9F39582ED296E">Representations for Associative Words</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
Reference Manual)
of the given <var>elm</var> relative to the generators the group was created
with or the free generators as returned by <code>FreeGeneratorsOfGroup</code>.
<p>
Continuing the above example:
<p>
<pre>
gap&gt; AsWordLetterRepInGenerators( f.1^2, u );    
[ 2, -1, 2 ]
gap&gt; AsWordLetterRepInFreeGenerators( f.1^2, u );
[ 2 ]
</pre>
<p>
This means: to get <code>f.1^2</code>, multiply the second of the given generators
with the inverse of the first and again with the second; or just take
the second free generator.
<p>
<p>
<h2><a name="SECT004">2.4 Automorphism groups of free groups</a></h2>
<p><p>
The <font face="Gill Sans,Helvetica,Arial">FGA</font> package knows presentations of the automorphism groups of free
groups. It also allows to express an automorphism as word in the
generators of these presentations.
This sections repeats the <font face="Gill Sans,Helvetica,Arial">GAP</font> standard methods to do so and shows
functions to obtain the generating automorphisms.
<p>
<a name = "SSEC004.1"></a>
<li><code>AutomorphismGroup( </code><var>u</var><code> ) A</code>
<p>
returns the automorphism group of the finitely generated subgroup <var>u</var>
of a free group.
<p>
Only a few methods will work with this group. But there is a way to
obtain an isomorphic finitely presented group:
<p>
<a name = "SSEC004.2"></a>
<li><code>IsomorphismFpGroup( </code><var>group</var><code> ) A</code>
<p>
returns an isomorphism of <var>group</var> to a finitely presented group.  
For automorphism groups of free groups, the <font face="Gill Sans,Helvetica,Arial">FGA</font> package implements
the presentations of <a href="biblio.htm#Neumann33"><cite>Neumann33</cite></a>.
The finitely presented group itself can then be obtained with the
command <code>Range</code>.
<p>
Here is an example:
<p>
<pre>
gap&gt; f := FreeGroup( 2 );;
gap&gt; a := AutomorphismGroup( f );;
gap&gt; iso := IsomorphismFpGroup( a );;
gap&gt; Range( iso );
&lt;fp group on the generators [ O, P, U ]&gt;
</pre>
<p>
To express an automorphism as word in the generators of the
presentation, just apply the isomorphism obtained from
<code>IsomorphismFpGroup</code>.
<p>
<pre>
gap&gt; aut := GroupHomomorphismByImages( f, f,
&gt;              GeneratorsOfGroup( f ), [ f.1^f.2, f.1*f.2 ] );
[ f1, f2 ] -&gt; [ f2^-1*f1*f2, f1*f2 ]
gap&gt; ImageElm( iso, aut );
O^2*U*O*P^-1*U
</pre>
<p>
It is also possible to use <code>aut^iso</code> or <code>Image( iso, aut )</code>.
Using <code>Image</code> will perform additional checks on the arguments.
<p>
The <font face="Gill Sans,Helvetica,Arial">FGA</font> package provides a simpler way to create endomorphisms:
<p>
<a name = "SSEC004.3"></a>
<li><code>FreeGroupEndomorphismByImages( </code><var>g</var><code>, </code><var>imgs</var><code> ) F</code>
<p>
returns the endomorphism that maps the free generators of the finitely
generated subgroup <var>g</var> of a free group to the elements listed in <var>imgs</var>.
You may then apply <code>IsBijective</code> to check whether it is an
automorphism.
<p>
The follwowing functions return automorphisms that correspond to the
generators in the presentation:
<p>
<a name = "SSEC004.4"></a>
<li><code>FreeGroupAutomorphismsGeneratorO( </code><var>group</var><code> ) F</code>
<a name = "SSEC004.4"></a>
<li><code>FreeGroupAutomorphismsGeneratorP( </code><var>group</var><code> ) F</code>
<a name = "SSEC004.4"></a>
<li><code>FreeGroupAutomorphismsGeneratorU( </code><var>group</var><code> ) F</code>
<a name = "SSEC004.4"></a>
<li><code>FreeGroupAutomorphismsGeneratorS( </code><var>group</var><code> ) F</code>
<a name = "SSEC004.4"></a>
<li><code>FreeGroupAutomorphismsGeneratorT( </code><var>group</var><code> ) F</code>
<a name = "SSEC004.4"></a>
<li><code>FreeGroupAutomorphismsGeneratorQ( </code><var>group</var><code> ) F</code>
<a name = "SSEC004.4"></a>
<li><code>FreeGroupAutomorphismsGeneratorR( </code><var>group</var><code> ) F</code>
<p>
return the automorphism which maps the free generators 
[ <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ..., <i>x</i><sub><i>n</i></sub> ] of <var>group</var> to
<dl compact>
  <dt>O:<dd>[ <i>x</i><sub>1</sub><sup>&#8722;1</sup>, <i>x</i><sub>2</sub>, ..., <i>x</i><sub><i>n</i></sub> ]                      (<i>n</i> &#8805; 1)
  <dt>P:<dd>[ <i>x</i><sub>2</sub>, <i>x</i><sub>1</sub>, <i>x</i><sub>3</sub>, ..., <i>x</i><sub><i>n</i></sub> ]                      (<i>n</i> &#8805; 2)
  <dt>U:<dd>[ <i>x</i><sub>1</sub><i>x</i><sub>2</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, ..., <i>x</i><sub><i>n</i></sub> ]                   (<i>n</i> &#8805; 2)
  <dt>S:<dd>[ <i>x</i><sub>2</sub><sup>&#8722;1</sup>, <i>x</i><sub>3</sub><sup>&#8722;1</sup>, ..., <i>x</i><sub><i>n</i></sub><sup>&#8722;1</sup>, <i>x</i><sub>1</sub><sup>&#8722;1</sup> ]  (<i>n</i> &#8805; 1)
  <dt>T:<dd>[ <i>x</i><sub>2</sub>, <i>x</i><sub>1</sub><sup>&#8722;1</sup>, <i>x</i><sub>3</sub>, ..., <i>x</i><sub><i>n</i></sub> ]                 (<i>n</i> &#8805; 2)
  <dt>Q:<dd>[ <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, ..., <i>x</i><sub><i>n</i></sub>, <i>x</i><sub>1</sub> ]                      (<i>n</i> &#8805; 2)
  <dt>R:<dd>[ <i>x</i><sub>2</sub><sup>&#8722;1</sup>, <i>x</i><sub>1</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, ..., <i>x</i><sub><i>n</i>&#8722;2</sub>, <i>x</i><sub><i>n</i></sub><i>x</i><sub><i>n</i>&#8722;1</sub><sup>&#8722;1</sup>, <i>x</i><sub><i>n</i>&#8722;1</sub><sup>&#8722;1</sup> ]           (<i>n</i> &#8805; 4)
</dl>
<p>
<p>
[<a href = "chapters.htm">Up</a>] [<a href ="CHAP001.htm">Previous</a>] [<a href ="CHAP003.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
<P>
<address>FGA manual<br>März 2018
</address></body></html>