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##
#W example4.tst KBMag Package Derek Holt
##
gap> START_TEST( "KBMag package: example4.tst" );
gap> fsa_infolevel_saved := InfoLevel( InfoFSA );;
gap> SetInfoLevel( InfoFSA, 0 );;
gap> rws_infolevel_saved := InfoLevel( InfoRWS );;
gap> SetInfoLevel( InfoRWS, 0 );;
gap> ## SubSection 1.9, Example 4
gap> ## This is an example of the use of the Knuth-Bendix algorithm
gap> ## to prove the nilpotence of a finitely presented group.
gap> ## (The method is due to Sims, described in Chapter 11.8 of [Sim94].)
gap> ## This example is of intermediate difficulty, and demonstrates the
gap> ## necessity of using the maxstoredlen control parameter.
gap> ## The group is ⟨a,b|[b,a,b],[b,a,a,a,a],[b,a,a,a,b,a,a]⟩
gap> ## with left-normed commutators. The first step in the method is to
gap> ## check that there is a maximal nilpotent quotient of the group,
gap> ## for which we could use, for example, the GAP NilpotentQuotient command,
gap> ## from the package “nq”. We find that there is a maximal such quotient,
gap> ## and it has class 7, and the layers going down the lower central series
gap> ## have the abelian structures [0,0], [0], [0], [0], [0], [2], [2].
gap> ## By using the stand-alone C nilpotent quotient program, it is possible
gap> ## to find a power-commutator presentation of this maximal quotient.
gap> ## We now construct a new presentation of the same group, by introducing
gap> ## the generators in this power-commutator presentation, together with
gap> ## their definitions as powers or commutators of earlier generators.
gap> ## This new presentation that we use as input for the Knuth-Bendix program.
gap> ## Again we use the recursive ordering, but this time we will be careful
gap> ## to introduce the generators in the correct order in the first place!
gap> ##
gap> F := FreeGroup( "h", "g", "f", "e", "d", "c", "b", "a" );;
gap> h:=F.1;; g:=F.2;; f:=F.3;; e:=F.4;;
gap> d:=F.5;; c:=F.6;; b:=F.7;; a:=F.8;;
gap> G := F/[Comm(b,a)*c^-1, Comm(c,a)*d^-1, Comm(d,a)*e^-1,
> Comm(e,b)*f^-1, Comm(f,a)*g^-1, Comm(g,b)*h^-1,
> Comm(g,a), Comm(c,b), Comm(e,a)];;
gap> R := KBMAGRewritingSystem( G );
rec(
isRWS := true,
generatorOrder := [_g1,_g2,_g3,_g4,_g5,_g6,_g7,_g8,_g9,_g10,_g11,_g12,_g13,
_g14,_g15,_g16],
inverses := [_g2,_g1,_g4,_g3,_g6,_g5,_g8,_g7,_g10,_g9,_g12,_g11,_g14,
_g13,_g16,_g15],
ordering := "shortlex",
equations := [
[_g14*_g16*_g13,_g11*_g16],
[_g12*_g16*_g11,_g9*_g16],
[_g10*_g16*_g9,_g7*_g16],
[_g8*_g14*_g7,_g5*_g14],
[_g6*_g16*_g5,_g3*_g16],
[_g4*_g14*_g3,_g1*_g14],
[_g4*_g16,_g16*_g4],
[_g12*_g14,_g14*_g12],
[_g8*_g16,_g16*_g8]
]
)
gap> SetOrderingOfKBMAGRewritingSystem( R, "recursive" );
gap> ## A little experimentation reveals that this example works best
gap> ## when only those equations with left and right hand sides of
gap> ## lengths at most 10 are kept.
gap> O := OptionsRecordOfKBMAGRewritingSystem( R );;
gap> O.maxstoredlen := [10,10];;
gap> SetInfoLevel( InfoRWS, 2 );
gap> KnuthBendix( R );
false
gap> ## Now it is essential to re-run with the ‘maxstoredlen’ limit removed
gap> ## to check that the system really is confluent.
gap> Unbind( O.maxstoredlen );
gap> KnuthBendix( R );
#I External Knuth-Bendix program complete.
#I System computed is confluent.
true
gap> ## #In fact, in this case, we did have a confluent set already.
gap> ## Inspection of the confluent set now reveals it to be precisely
gap> ## a power-commutator presentation of a nilpotent group, and so we
gap> ## have proved that the group we started with really is nilpotent.
gap> ## Of course, this means also that it is equal to its largest
gap> ## nilpotent quotient, of which we already know the structure.
gap> ## SetInfoLevel( InfoFSA, fsa_infolevel_saved );;
gap> ## SetInfoLevel( InfoRWS, rws_infolevel_saved );;
gap> STOP_TEST( "example4.tst", 10000 );
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2026-03-28
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