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> f := FreeGroup( 2 );
<free group on the generators [ f1, f2 ]>
gap> u := Group( f.1^2, f.2^2, f.1*f.2 );
Group([ f1^2, f2^2, f1*f2 ])
gap> u1 := Subgroup( u, [f.1^2, f.1^4*f.2^6] );
Group([ f1^2, f1^4*f2^6 ])
gap> elm := f.1;
f1
gap> 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 ≠ 1 implies |<i>v</i>1 <i>v</i>2 | ≥ max(|<i>v</i>1 |, |<i>v</i>2 |), and
<li>
<i>v</i>1 <i>v</i>2 ≠ 1 and <i>v</i>2 <i>v</i>3 ≠ 1 implies
|<i>v</i>1 <i>v</i>2 <i>v</i>3 | > |<i>v</i>1 | − |<i>v</i>2 | + |<i>v</i>3 |
</ul>
<p>
hold, where |·| 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 〈<i>elm</i> 〉
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> ∈ <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> f := FreeGroup( 2 );
<free group on the generators [ f1, f2 ]>
gap> ua := f.1^2*f.2^2;; ub := f.1^2*f.2;;
gap> u := Group( ua, ub );;
gap> g := FreeGroup( "a", "b" );;
gap> hom := GroupHomomorphismByImages( g, u,
> GeneratorsOfGroup(g),
> GeneratorsOfGroup(u) );
[ a, b ] -> [ f1^2*f2^2, f1^2*f2 ]
gap> # how can f.1^2 be expressed?
gap> PreImagesRepresentative( hom, f.1^2 );
b*a^-1*b
gap> last ^ hom; # check this
f1^2
gap> ub * ua^-1 * ub; # another check
f1^2
gap> PreImagesRepresentative( hom, f.1 ); # try f.1
fail
gap> 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 <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> AsWordLetterRepInGenerators( f.1^2, u );
[ 2, -1, 2 ]
gap> 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> f := FreeGroup( 2 );;
gap> a := AutomorphismGroup( f );;
gap> iso := IsomorphismFpGroup( a );;
gap> Range( iso );
<fp group on the generators [ O, P, U ]>
</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> aut := GroupHomomorphismByImages( f, f,
> GeneratorsOfGroup( f ), [ f.1^f.2, f.1*f.2 ] );
[ f1, f2 ] -> [ f2^-1*f1*f2, f1*f2 ]
gap> 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>−1</sup>, <i>x</i><sub>2</sub>, ..., <i>x</i><sub><i>n</i></sub> ] (<i>n</i> ≥ 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> ≥ 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> ≥ 2)
<dt>S:<dd>[ <i>x</i><sub>2</sub><sup>−1</sup>, <i>x</i><sub>3</sub><sup>−1</sup>, ..., <i>x</i><sub><i>n</i></sub><sup>−1</sup>, <i>x</i><sub>1</sub><sup>−1</sup> ] (<i>n</i> ≥ 1)
<dt>T:<dd>[ <i>x</i><sub>2</sub>, <i>x</i><sub>1</sub><sup>−1</sup>, <i>x</i><sub>3</sub>, ..., <i>x</i><sub><i>n</i></sub> ] (<i>n</i> ≥ 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> ≥ 2)
<dt>R:<dd>[ <i>x</i><sub>2</sub><sup>−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>−2</sub>, <i>x</i><sub><i>n</i></sub><i>x</i><sub><i>n</i>−1</sub><sup>−1</sup>, <i>x</i><sub><i>n</i>−1</sub><sup>−1</sup> ] (<i>n</i> ≥ 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>
|