1<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 2<!-- %% --> 3<!-- %A grpfp.msk GAP documentation Alexander Hulpke --> 4<!-- %% Volkmar Felsch --> 5<!-- %% --> 6<!-- %A @(#)<M>Id: grpfp.msk,v 1.84 2006/03/08 14:50:04 jjm Exp </M> --> 7<!-- %% --> 8<!-- %Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland --> 9<!-- %Y Copyright (C) 2002 The GAP Group --> 10<!-- %% --> 11<Chapter Label="Finitely Presented Groups"> 12<Heading>Finitely Presented Groups</Heading> 13 14A <E>finitely presented group</E> (in short: FpGroup) is a group generated by 15a finite set of <E>abstract generators</E> subject to a finite set of 16<E>relations</E> that these generators satisfy. 17Every finite group can be represented as a finitely presented group, 18though in almost all cases it is computationally much more efficient to work 19in another representation (even the regular permutation representation). 20<P/> 21Finitely presented groups are obtained by factoring a free group by a set 22of relators. Their elements know about this presentation and compare 23accordingly. 24<P/> 25So to create a finitely presented group you first have to generate a free 26group (see <Ref Func="FreeGroup" Label="for given rank"/> for details). 27 28There are two ways to specify a quotient of the free group: either by giving 29a list of relators or by giving a list of equations. 30 31Relators are just words in the generators of the free group. Equations are 32represented as pairs of words in the generators of the free group. 33 34In either case the generators of the quotient are <E>the images</E> of the free 35generators under the canonical homomorphism from the free group onto the quotient. 36So for example to create the group 37<Display Mode="M"> 38\langle a, b \mid a^2, b^3, (a b)^5 \rangle 39</Display> 40you can use the following commands: 41<Example><![CDATA[ 42gap> f := FreeGroup( "a", "b" );; 43gap> g := f / [ f.1^2, f.2^3, (f.1*f.2)^5 ]; 44<fp group on the generators [ a, b ]> 45gap> h := f / [ [f.1^2, f.1^0], [f.2^3, f.1^0], [(f.1*f.2)^4, f.2^-1*f.1^-1] ]; 46<fp group on the generators [ a, b ]> 47]]></Example> 48<P/> 49Note that you cannot call the generators by their names. These names are 50not variables, but just display figures. So, if you want to access the 51generators by their names, you first have to introduce the respective 52variables and to assign the generators to them. 53<P/> 54<Example><![CDATA[ 55gap> Unbind(a); 56gap> GeneratorsOfGroup( g ); 57[ a, b ] 58gap> a; 59Error, Variable: 'a' must have a value 60gap> a := g.1;; b := g.2;; # assign variables 61gap> GeneratorsOfGroup( g ); 62[ a, b ] 63gap> a in f; 64false 65gap> a in g; 66true 67]]></Example> 68<P/> 69To relieve you of the tedium of typing the above assignments, 70<E>when working interactively</E>, 71there is the function <Ref Oper="AssignGeneratorVariables"/>. 72<P/> 73Note that the generators of the free group are different from the 74generators of the FpGroup (even though they are displayed by the same 75names). That means that words in the generators of the free group are not 76elements of the finitely presented group. Vice versa elements of the 77FpGroup are not words. 78<P/> 79<Example><![CDATA[ 80gap> a*b = b*a; 81false 82gap> (b^2*a*b)^2 = a^0; 83true 84]]></Example> 85<P/> 86Such calculations comparing elements of an FpGroup may run into problems: 87There exist finitely 88presented groups for which no algorithm exists (it is known that no such 89algorithm can exist) that will tell for two arbitrary words in the 90generators whether the corresponding elements in the FpGroup are equal. 91<P/> 92Therefore the methods used by &GAP; to compute in finitely 93presented groups may run into warning errors, run out of memory or run 94forever. If the FpGroup is (by theory) known to be finite the 95algorithms are guaranteed to terminate (if there is sufficient memory 96available), but the time needed for the calculation cannot be bounded a 97priori. See <Ref Sect="Coset Tables and Coset Enumeration"/> and 98<Ref Sect="Testing Finiteness of Finitely Presented Groups"/>. 99<P/> 100<Example><![CDATA[ 101gap> (b^2*a*b)^2; 102(b^2*a*b)^2 103gap> a^0; 104<identity ...> 105]]></Example> 106<P/> 107A consequence of our convention is that elements of finitely presented 108groups are not printed in a unique way. 109See also <Ref Func="SetReducedMultiplication"/>. 110 111 112<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 113<Section Label="sect:IsSubgroupFpGroup"> 114<Heading>IsSubgroupFpGroup and IsFpGroup</Heading> 115 116<#Include Label="IsSubgroupFpGroup"> 117<#Include Label="IsFpGroup"> 118<#Include Label="InfoFpGroup"> 119 120</Section> 121 122 123<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 124<Section Label="Creating Finitely Presented Groups"> 125<Heading>Creating Finitely Presented Groups</Heading> 126 127<ManSection> 128<Meth Name="\/" Arg="F, rels" 129 Label="for a free group and a list of elements"/> 130<Meth Name="\/" Arg="F, eqns" 131 Label="for a free group and a list of pairs of elements"/> 132 133<Description> 134<Index Subkey="for finitely presented groups">quotient</Index> 135creates a finitely presented group given by the presentation 136<M>\langle gens \mid <A>rels</A> \rangle</M> or 137<M>\langle gens \mid <A>eqns</A> \rangle</M>, respectively 138where <M>gens</M> are the free generators of the free group <A>F</A>. 139Relations can be entered either as words or 140as pairs of words in the generators of <A>F</A>. In the former case 141we refer to the words given as <E>relators</E>, in the latter we 142refer to the pairs of words as <E>equations</E>. 143The two methods can currently not be mixed. 144<P/> 145The same result is obtained with the infix operator <C>/</C>, 146i.e., as <A>F</A> <C>/</C> <A>rels</A>. 147<P/> 148<Example><![CDATA[ 149gap> f := FreeGroup( 3 );; 150gap> f / [ f.1^4, f.2^3, f.3^5, f.1*f.2*f.3 ]; 151<fp group on the generators [ f1, f2, f3 ]> 152gap> f / [ [ f.1^4, f.1^0 ], [ f.2^3, f.1^0 ], [ f.1, f.2^-1*f.3^-1 ] ]; 153<fp group on the generators [ f1, f2, f3 ]> 154]]></Example> 155</Description> 156</ManSection> 157 158<#Include Label="FactorGroupFpGroupByRels"> 159<#Include Label="ParseRelators"> 160<#Include Label="StringFactorizationWord"> 161 162</Section> 163 164 165<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 166<Section Label="Comparison of Elements of Finitely Presented Groups"> 167<Heading>Comparison of Elements of Finitely Presented Groups</Heading> 168 169<ManSection> 170<Meth Name="\=" Arg="a, b" Label="for two elements in a f.p. group"/> 171 172<Description> 173<Index Subkey="elements of finitely presented groups">equality</Index> 174Two elements of a finitely presented group are equal if they are equal in 175this group. Nevertheless they may be represented as different words in the 176generators. Because of the fundamental problems mentioned in the 177introduction to this chapter such a test may take very long and cannot be 178guaranteed to finish. 179<P/> 180The method employed by &GAP; for such an equality test use the underlying 181finitely presented group. First (unless this group is known to be infinite) 182&GAP; tries to find a faithful permutation representation by a bounded 183Todd-Coxeter. 184If this fails, a Knuth-Bendix 185(see <Ref Sect="Rewriting Systems and the Knuth-Bendix Procedure"/>) 186is attempted and the words are compared via their normal form. 187<P/> 188If only elements in a subgroup are to be tested for equality it thus can be 189useful to translate the problem in a new finitely presented group by 190rewriting (see <Ref Attr="IsomorphismFpGroup"/>); 191<P/> 192The equality test of elements underlies many <Q>basic</Q> calculations, 193such as the order of an element, 194and the same type of problems can arise there. 195In some cases, working with rewriting systems can still help to solve the 196problem. 197The <Package>kbmag</Package> package provides such functionality, 198see the package manual for further details. 199</Description> 200</ManSection> 201 202 203<ManSection> 204<Meth Name="\<" Arg="a, b" Label="for two elements in a f.p. group"/> 205 206<Description> 207<Index Subkey="elements of finitely presented groups">smaller</Index> 208Compared with equality testing, 209problems get even worse when trying to compute a total ordering on the 210elements of a finitely presented group. As any ordering that is guaranteed 211to be reproducible in different runs of &GAP; or even with different groups 212given by syntactically equal presentations would be prohibitively expensive 213to implement, the ordering of elements is depending on a method chosen by 214&GAP; and not guaranteed to stay the same when repeating the construction 215of an FpGroup. The only guarantee given for the <C><</C> 216ordering for such elements is that it will stay the same for one family 217during its lifetime. 218The attribute <Ref Attr="FpElmComparisonMethod"/> is used to obtain 219a comparison function for a family of FpGroup elements. 220</Description> 221</ManSection> 222 223<#Include Label="FpElmComparisonMethod"> 224<#Include Label="SetReducedMultiplication"> 225 226</Section> 227 228 229<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 230<Section Label="Preimages in the Free Group"> 231<Heading>Preimages in the Free Group</Heading> 232 233<#Include Label="FreeGroupOfFpGroup"> 234<#Include Label="FreeGeneratorsOfFpGroup"> 235<#Include Label="RelatorsOfFpGroup"> 236 237 238<ManSection> 239<Oper Name="UnderlyingElement" Arg='elm' Label="fp group elements"/> 240 241<Description> 242Let <A>elm</A> be an element of a group whose elements are represented as 243words with further properties. 244Then <Ref Oper="UnderlyingElement" Label="fp group elements"/> returns 245the word from the free group that is used as a representative for <A>elm</A>. 246<P/> 247<Example><![CDATA[ 248gap> w := g.1*g.2; 249a*b 250gap> IsWord( w ); 251false 252gap> ue := UnderlyingElement( w ); 253a*b 254gap> IsWord( ue ); 255true 256]]></Example> 257</Description> 258</ManSection> 259 260 261<#Include Label="ElementOfFpGroup"> 262 263</Section> 264 265 266<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 267<Section Label="Operations for Finitely Presented Groups"> 268<Heading>Operations for Finitely Presented Groups</Heading> 269 270Finitely presented groups are groups and so all operations for groups should 271be applicable to them (though not necessarily efficient methods are 272available). 273Most methods for finitely presented groups rely on coset enumeration. 274See <Ref Sect="Coset Tables and Coset Enumeration"/> for details. 275<P/> 276The command <Ref Attr="IsomorphismPermGroup"/> can be used to obtain 277a faithful permutation representation, 278if such a representation of small degree exists. 279(Otherwise it might run very long or fail.) 280<Example><![CDATA[ 281gap> f := FreeGroup( "a", "b" ); 282<free group on the generators [ a, b ]> 283gap> g := f / [ f.1^2, f.2^3, (f.1*f.2)^5 ]; 284<fp group on the generators [ a, b ]> 285gap> h := IsomorphismPermGroup( g ); 286[ a, b ] -> [ (1,2)(3,5), (2,3,4) ] 287gap> u:=Subgroup(g,[g.1*g.2]);;rt:=RightTransversal(g,u); 288RightTransversal(<fp group of size 60 on the generators 289[ a, b ]>,Group([ a*b ])) 290gap> Image(ActionHomomorphism(g,rt,OnRight)); 291Group([ (1,2)(3,4)(5,7)(6,8)(9,10)(11,12), 292 (1,3,2)(4,5,6)(7,8,9)(10,11,12) ]) 293]]></Example> 294 295 296<ManSection> 297<Meth Name="PseudoRandom" Arg='F:radius:=l' 298 Label="for finitely presented groups"/> 299 300<Description> 301The default algorithm for <Ref Oper="PseudoRandom"/> 302makes little sense for finitely presented or free groups, 303as it produces words that are extremely long. 304<P/> 305By specifying the option <C>radius</C>, 306instead elements are taken as words in the generators of <A>F</A> 307in the ball of radius <A>l</A> with equal distribution in the free group. 308<P/> 309<Log><![CDATA[ 310gap> PseudoRandom(g:radius:=20); 311a^3*b^2*a^-2*b^-1*a*b^-4*a*b^-1*a*b^-4 312]]></Log> 313</Description> 314</ManSection> 315 316</Section> 317 318 319<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 320<Section Label="Coset Tables and Coset Enumeration"> 321<Heading>Coset Tables and Coset Enumeration</Heading> 322 323Coset enumeration (see <Cite Key="Neu82"/> for an explanation) is one of the 324fundamental tools for the examination of finitely presented groups. 325This section describes &GAP; functions that can be used to invoke a coset 326enumeration. 327<P/> 328Note that in addition to the built-in coset enumerator there is the &GAP; 329package <Package>ACE</Package>. 330Moreover, &GAP; provides an interactive Todd-Coxeter 331in the &GAP; package <Package>ITC</Package> 332which is based on the <Package>XGAP</Package> package. 333 334<#Include Label="CosetTable"> 335<#Include Label="TracedCosetFpGroup"> 336 337<ManSection> 338<Oper Name="FactorCosetAction" Arg='G, H' Label="for fp groups"/> 339 340<Description> 341returns the action of <A>G</A> on the cosets of its subgroup <A>H</A>. 342<P/> 343<Example><![CDATA[ 344gap> u := Subgroup( g, [ g.1, g.1^g.2 ] ); 345Group([ a, b^-1*a*b ]) 346gap> FactorCosetAction( g, u ); 347[ a, b ] -> [ (2,4)(5,6), (1,2,3)(4,5,6) ] 348]]></Example> 349</Description> 350</ManSection> 351 352 353<#Include Label="CosetTableBySubgroup"> 354<#Include Label="CosetTableFromGensAndRels"> 355<#Include Label="CosetTableDefaultMaxLimit"> 356<#Include Label="CosetTableDefaultLimit"> 357<#Include Label="MostFrequentGeneratorFpGroup"> 358<#Include Label="IndicesInvolutaryGenerators"> 359 360</Section> 361 362 363<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 364<Section Label="Standardization of coset tables"> 365<Heading>Standardization of coset tables</Heading> 366 367For any two coset numbers <M>i</M> and <M>j</M> with <M>i < j</M> 368the first occurrence of <M>i</M> in a coset table precedes 369the first occurrence of <M>j</M> with respect to 370the usual row-wise ordering of the table entries. Following the notation of 371Charles Sims' book on computation with finitely presented groups 372<Cite Key="Sims94"/> we call such a table a <E>standard coset table</E>. 373<P/> 374The table entries which contain the first occurrences of the coset numbers 375<M>i > 1</M> recursively provide for each <M>i</M> a representative of the 376corresponding coset in form of a unique word <M>w_i</M> in the generators and 377inverse generators of <M>G</M>. 378The first coset (which is <M>H</M> itself) can be represented by 379the empty word <M>w_1</M>. A coset table is standard if and only 380if the words <M>w_1, w_2, \ldots</M> are length-plus-lexicographic ordered 381(as defined in <Cite Key="Sims94"/>), for short: <E>lenlex</E>. 382<P/> 383This standardization of coset tables is 384different from that used in &GAP; versions 4.2 and earlier. Before 385that, we ignored the columns that correspond to inverse generators and 386hence only considered words in the generators of <M>G</M>. We call 387this older ordering the <E>semilenlex</E> standard as it 388also applies to the case of semigroups where no inverses of the generators are known. 389<P/> 390We changed our default from the semilenlex standard to the lenlex 391standard to be consistent with <Cite Key="Sims94"/>. However, the 392semilenlex standardisation remains available and the convention used 393for all implicit standardisations can be selected by setting the value of the global variable 394<Ref Var="CosetTableStandard"/> to either <C>"lenlex"</C> or 395<C>"semilenlex"</C>. 396 397Independent of the current value of <Ref Var="CosetTableStandard"/> 398you can standardize (or restandardize) a coset table at any 399time using <Ref Func="StandardizeTable"/>. 400 401<#Include Label="CosetTableStandard"> 402<#Include Label="StandardizeTable"> 403 404</Section> 405 406 407<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 408<Section Label="Coset tables for subgroups in the whole group"> 409<Heading>Coset tables for subgroups in the whole group</Heading> 410 411<#Include Label="CosetTableInWholeGroup"> 412<#Include Label="SubgroupOfWholeGroupByCosetTable"> 413 414</Section> 415 416 417<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 418<Section Label="Augmented Coset Tables and Rewriting"> 419<Heading>Augmented Coset Tables and Rewriting</Heading> 420 421<#Include Label="AugmentedCosetTableInWholeGroup"> 422<#Include Label="AugmentedCosetTableMtc"> 423<#Include Label="AugmentedCosetTableRrs"> 424<#Include Label="RewriteWord"> 425 426</Section> 427 428 429<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 430<Section Label="Low Index Subgroups"> 431<Heading>Low Index Subgroups</Heading> 432 433<#Include Label="LowIndexSubgroupsFpGroupIterator"> 434 435</Section> 436 437 438<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 439<Section Label="Converting Groups to Finitely Presented Groups"> 440<Heading>Converting Groups to Finitely Presented Groups</Heading> 441 442<#Include Label="IsomorphismFpGroup"> 443<#Include Label="IsomorphismFpGroupByGenerators"> 444 445</Section> 446 447 448<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 449<Section Label="New Presentations and Presentations for Subgroups"> 450<Heading>New Presentations and Presentations for Subgroups</Heading> 451 452<Index Key="IsomorphismFpGroup" Subkey="for subgroups of fp groups"> 453<C>IsomorphismFpGroup</C></Index> 454<Ref Attr="IsomorphismFpGroup"/> is also used to compute 455a new finitely presented group that is isomorphic to the given subgroup 456of a finitely presented group. 457(This is typically the only method to compute with subgroups of a finitely 458presented group.) 459<P/> 460<Example><![CDATA[ 461gap> f:=FreeGroup(2);; 462gap> g:=f/[f.1^2,f.2^3,(f.1*f.2)^5]; 463<fp group on the generators [ f1, f2 ]> 464gap> u:=Subgroup(g,[g.1*g.2]); 465Group([ f1*f2 ]) 466gap> hom:=IsomorphismFpGroup(u); 467[ <[ [ 1, 1 ] ]|f2^-1*f1^-1> ] -> [ F1 ] 468gap> new:=Range(hom); 469<fp group on the generators [ F1 ]> 470gap> List(GeneratorsOfGroup(new),i->PreImagesRepresentative(hom,i)); 471[ <[ [ 1, 1 ] ]|f2^-1*f1^-1> ] 472]]></Example> 473<P/> 474When working with such homomorphisms, some subgroup elements are expressed 475as extremely long words in the group generators. Therefore the underlying 476words of subgroup 477generators stored in the isomorphism (as obtained by 478<Ref Attr="MappingGeneratorsImages"/> and displayed when 479<Ref Func="View"/>ing the homomorphism) 480as well as preimages under the homomorphism are stored in the form of 481straight line program elements 482(see <Ref Sect="Straight Line Program Elements"/>). These will 483behave like ordinary words and no extra treatment should be necessary. 484<P/> 485<Example><![CDATA[ 486gap> r:=Range(hom).1^10; 487F1^10 488gap> p:=PreImagesRepresentative(hom,r); 489<[ [ 1, 10 ] ]|(f2^-1*f1^-1)^10> 490]]></Example> 491 492If desired, it also is possible to convert these underlying words using 493<Ref Func="EvalStraightLineProgElm"/>: 494 495<Example><![CDATA[ 496gap> r:=EvalStraightLineProgElm(UnderlyingElement(p)); 497(f2^-1*f1^-1)^10 498gap> p:=ElementOfFpGroup(FamilyObj(p),r); 499(f2^-1*f1^-1)^10 500]]></Example> 501<P/> 502(If you are only interested in a finitely presented group isomorphic to 503the given subgroup but not in the isomorphism, 504you may also use the functions 505<Ref Func="PresentationViaCosetTable"/> and <Ref Func="FpGroupPresentation"/> 506(see <Ref Sect="Creating Presentations"/>).) 507<P/> 508Homomorphisms can also be used to obtain an isomorphic finitely presented 509group with a (hopefully) simpler presentation. 510<P/> 511<#Include Label="IsomorphismSimplifiedFpGroup"> 512 513</Section> 514 515 516<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 517<Section Label="Preimages under Homomorphisms from an FpGroup"> 518<Heading>Preimages under Homomorphisms from an FpGroup</Heading> 519 520For some subgroups of a finitely presented group the number of 521subgroup generators increases with the index of the subgroup. However often 522these generators are not needed at all for further calculations, but what is 523needed is the action of the cosets of the subgroup. This gives the image of 524the subgroup in a finite quotient and this finite quotient can be used to 525calculate normalizers, closures, intersections and so 526forth <Cite Key="HulpkeQuot"/>. 527<P/> 528The same applies for subgroups that are obtained as preimages under 529homomorphisms. 530 531<#Include Label="SubgroupOfWholeGroupByQuotientSubgroup"> 532<#Include Label="IsSubgroupOfWholeGroupByQuotientRep"> 533<#Include Label="AsSubgroupOfWholeGroupByQuotient"> 534<#Include Label="DefiningQuotientHomomorphism"> 535 536</Section> 537 538 539<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 540<Section Label="Quotient Methods"> 541<Heading>Quotient Methods</Heading> 542 543An important class of algorithms for finitely presented groups are the 544<E>quotient algorithms</E> which compute quotient groups of a given finitely 545presented group. There are algorithms for epimorphisms onto abelian groups, 546<M>p</M>-groups and solvable groups. 547(The <Q>low index</Q> algorithm 548–<Ref Oper="LowIndexSubgroupsFpGroup"/>– 549can be considered as well as an algorithm that produces permutation group 550quotients.) 551<P/> 552<Ref Attr="MaximalAbelianQuotient"/>, 553as defined for general groups, returns the largest abelian 554quotient of the given group. 555 556<Example><![CDATA[ 557gap> f:=FreeGroup(2);;fp:=f/[f.1^6,f.2^6,(f.1*f.2)^12]; 558<fp group on the generators [ f1, f2 ]> 559gap> hom:=MaximalAbelianQuotient(fp); 560[ f1, f2 ] -> [ f1, f3 ] 561gap> Size(Image(hom)); 56236 563]]></Example> 564 565<#Include Label="PQuotient"> 566<#Include Label="EpimorphismQuotientSystem"> 567<#Include Label="EpimorphismPGroup"> 568<#Include Label="EpimorphismNilpotentQuotient"> 569<#Include Label="SolvableQuotient"> 570<#Include Label="EpimorphismSolvableQuotient"> 571<#Include Label="LargerQuotientBySubgroupAbelianization"> 572 573</Section> 574 575 576<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 577<Section Label="Abelian Invariants for Subgroups"> 578<Heading>Abelian Invariants for Subgroups</Heading> 579 580Using variations of coset enumeration it is possible to compute the abelian 581invariants of a subgroup of a finitely presented group without computing a 582complete presentation for the subgroup in the first place. 583Typically, the operation <Ref Attr="AbelianInvariants"/> when called for 584subgroups should automatically take care of this, 585but in case you want to have further control about the methods used, 586the following operations might be of use. 587 588<#Include Label="AbelianInvariantsSubgroupFpGroup"> 589<#Include Label="AbelianInvariantsSubgroupFpGroupMtc"> 590<#Include Label="AbelianInvariantsSubgroupFpGroupRrs"> 591<#Include Label="AbelianInvariantsNormalClosureFpGroup"> 592<#Include Label="AbelianInvariantsNormalClosureFpGroupRrs"> 593 594</Section> 595 596 597<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 598<Section Label="Testing Finiteness of Finitely Presented Groups"> 599<Heading>Testing Finiteness of Finitely Presented Groups</Heading> 600 601As a consequence of the algorithmic insolvabilities mentioned in the 602introduction to this chapter, there cannot be a general method that will 603test whether a given finitely presented group is actually finite. 604<P/> 605Therefore testing the finiteness of a finitely presented group 606can be problematic. 607What &GAP; actually does upon a call of <Ref Prop="IsFinite"/> 608(or if it is –probably implicitly– asked for a faithful 609permutation representation) 610is to test whether it can find (via coset enumeration) a 611cyclic subgroup of finite index. If it can, it rewrites the presentation to 612this subgroup. Since the subgroup is cyclic, its size can be checked easily 613from the resulting presentation, the size of the whole group is the product 614of the index and the subgroup size. Since however no bound for the index of 615such a subgroup (if any exist) is known, such a test might continue 616unsuccessfully until memory is exhausted. 617<P/> 618On the other hand, a couple of methods exist, that might prove that a group 619is infinite. Again, none is guaranteed to work in every case: 620<P/> 621The first method is to find (for example via the low index algorithm, 622see <Ref Oper="LowIndexSubgroupsFpGroup"/>) a subgroup <M>U</M> 623such that <M>[U:U']</M> is infinite. 624If <M>U</M> has finite index, this can be checked by 625<Ref Prop="IsInfiniteAbelianizationGroup"/>. 626<P/> 627Note that this test has been done traditionally by checking the 628<Ref Attr="AbelianInvariants"/> 629(see section <Ref Sect="Abelian Invariants for Subgroups"/>) 630of <M>U</M>, 631<Ref Prop="IsInfiniteAbelianizationGroup"/> does a 632similar calculation but stops as soon as it is known whether <M>0</M> is an 633invariant without computing the actual values. This can be notably faster. 634<P/> 635Another method is based on <M>p</M>-group quotients, 636see <Ref Func="NewmanInfinityCriterion"/>. 637 638<#Include Label="IsInfiniteAbelianizationGroup:grp"> 639<#Include Label="NewmanInfinityCriterion"> 640 641</Section> 642</Chapter> 643 644