Theorem. Let G be a group with a finite presentation on b generators and r relations, and suppose p is an odd prime. Let d denote the rank of the elementary abelian group G1 = [G, G]Gp and let e denote the rank of G2 = [G1, G]Gp. If r - b < d2/2 - d/2 - d - e or r - b ≤d2/2 - d/2 - d - e + (e + d/2 - d2/4)d/2, then G has arbitrary large quotients of p-power order.
We present a proof that F(9) is infinite using this result.
> Left := func< b, r | r - b >;
> Right := func< d, e | d^2 div 2 - d div 2 - d - e +
> (e + d div 2 - d^2 div 4)*(d div 2) >;
>
>
> F< x1,x2,x3,x4,x5,x6,x7,x8,x9 > :=
> FPGroup< x1, x2, x3, x4, x5, x6, x7, x8, x9 |
> x1*x2=x3, x2*x3=x4, x3*x4=x5, x4*x5=x6, x5*x6=x7,
> x6*x7=x8, x7*x8=x9, x8*x9=x1, x9*x1=x2 >;
>
> F;
Finitely presented group F on 9 generators
Relations
x1 * x2 = x3
x2 * x3 = x4
x3 * x4 = x5
x4 * x5 = x6
x5 * x6 = x7
x6 * x7 = x8
x7 * x8 = x9
x8 * x9 = x1
x9 * x1 = x2
> AbelianQuotientInvariants(F);
[ 2, 38 ]
Thus the nilpotent quotient of F is divisible by 2 and 19.
We examine the 2- and 19-quotients of F.
> Q1 := pQuotient(F, 2, 0: Print := 1); Class limit set to 127. Lower exponent-2 central series for F Group: F to lower exponent-2 central class 1 has order 2^2 Group: F to lower exponent-2 central class 2 has order 2^3 Group completed. Lower exponent-2 central class = 2, Order = 2^3 > Q2 := pQuotient(F, 19, 0: Print := 1); Class limit set to 127. Lower exponent-19 central series for F Group: F to lower exponent-19 central class 1 has order 19^1 Group completed. Lower exponent-19 central class = 1, Order = 19^1Thus, the nilpotent residual of F has index 152. We try to locate this subgroup. We first take a 2-generator presentation for F.
> G := Simplify(F);
> G;
Finitely presented group G on 2 generators
Generators as words in group F
G.1 = x4
G.2 = x5
Relations
G.2 * G.1 * G.2 * G.1 * G.2^2 * G.1 * G.2^2 * G.1^-1 * G.2 *
G.1^-2 * G.2 * G.1^-2 = Id(G)
G.1 * G.2^2 * G.1 * G.2 * G.1^2 * G.2^-1 * G.1^2 * G.2^-1 *
G.1 * G.2^-1 * G.1^2 * G.2^-1 * G.1 * G.2^-1 = Id(G)
> H := ncl< G | (G.1, G.2) >;
> H;
Finitely presented group H
Index in group G is 76 = 2^2 * 19
Subgroup of group G defined by coset table
We haven't got the full nilpotent residual yet, so we try again.
> H := ncl< G | (G.1*G.1, G.2) >; > H; Finitely presented group H Index in group G is 152 = 2^3 * 19 Subgroup of group G defined by coset tableNow, we have it.
> AbelianQuotientInvariants(H); [ 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5 ]The nilpotent residual H has a 5-quotient. We construct a presentation for H and then calculate d and e for its 5-quotient.
> K := Rewrite(G, H: Simplify := false); > KP := pQuotientProcess(K, 5, 1); > d := FactoredOrder(ExtractGroup(KP))[1][2]; > NextClass(~KP); > e := FactoredOrder(ExtractGroup(KP))[1][2] - d; > "D = ", d, "e = ", e; d = 18 e = 81 > "Right hand side = ", Right(d, e); Right hand side = 135 > "Left hand side = ", Left(Ngens(K), #Relations(K)); Left hand side = 151Since Left is greater than Right, this presentation for H doesn't work. Thus we start eliminating generators.
> K := Simplify(K: Iterations := 1); > "Left hand side = ", Left(Ngens(K), #Relations(K)); Left hand side = 136 > K := Simplify(K: Iterations := 1); > "Left hand side = ", Left(Ngens(K), #Relations(K)); Left hand side = 123Got it! By Newman's theorem, H has an infinite 5-quotient and so F must be infinite.
We note in passing that the strategy outlined in this example is, together with other approaches, applied by the function Order.
> F<a,b,c> := FreeGroup(3); > rels := [ a^4, b^2, (a,b)^3, (a*b)^8, ((a*b)^2*(a^-1*b)^3)^2, > c^2, (c*a*c*a^2)^2, (c*a)^3*(c*a^-1)^3, > c*a*b*c*a^-1*b*a^-1*c*a*b*c*a^2*a*b*a^-1, > c*a*b*c*b*a*c*a*c*a^-1*b*c*b*a^-1*c*a^-1, > c*a*b*a^-1*c*a*b*a^-1*c*a*b*a^-1*c*a*b*a^-1, > c*b*a^2*b*c*b*c*a^2*c*b*c*b*a^2*b, > c*a^2*c*b*a*c*b*a*c*b*a*c*a^-1*c*a*b*a^-1, > c*a^-1*b*a*c*a^-1*b*a*c*b*a*b*a^2*b*a^-1*b, > c*a*b*a^-1*b*c*b*a^-1*b*c*a^-1*b*a*b*a*c*b*c*b, > c*a*c*b*a*b*c*a*c*b*a*b*c*a*c*b*a*b, > c*b*a^-1*b*c*a^-1*c*a^-1*b*a*b*c*b*c*a^2*b*a*b*a^-1, > c*b*a^-1*b*a*b*c*b*a^-1*b*a*b*c*b*a^-1*b*a*b, > c*a^2*b*a^-1*b*c*b*c*b*a^-1*b*a*c*b*a^2*b*a^-1*b > ]; > G<a,b,c> := quo< F | rels >;As it happens, trying to determine the order of G by enumerating the cosets of the trivial subgroup is quite hard. -- Even the predefined enumeration strategy "Hard" (cf. ToddCoxeter and CosetEnumerationProcess) does not give a finite result.
> time ToddCoxeter(G, sub<G|> : Strategy := "Hard"); 0 Time: 199.620Of course we could try to increase the workspace for the coset enumeration, but we decide to be more clever. Trying random words in the generators of G, we easily find some subgroup S of G with pretty small index in G.
> S := sub< G | b, c*a*c*b*a*b >; > time Index(G, S); 114 Time: 0.120We now have to compute the order of S. In order to be able to do this using coset enumeration, we have to construct a presentation for S by rewriting S w.r.t. G.
> time R := Rewrite(G, S : Simplify := false); Time: 0.030However, the presentation obtained by Reidemeister-Schreier rewriting without any simplification is not suitable for coset enumeration: it contains too many generators and its total length is quite high.
> NumberOfGenerators(R); 133 > PresentationLength(R); 14384An enumeration with the predefined enumeration strategy "Hard" does not produce a finite result. (Note that in consideration of the high total relator length, we select a coset table based enumeration style; cf. CosetEnumerationProcess.)
> time ToddCoxeter(R, sub<R|> : Strategy := "Hard", Style := "C"); 0 Time: 4.330On the other hand, simplifying the presentation by reducing the number of generators as much as possible is not a good idea either, since the total relator length grows massively.
> time Rs := Simplify(R); Time: 43.900 > NumberOfGenerators(Rs); 3 > PresentationLength(Rs); 797701Again, an enumeration with the predefined enumeration strategy "Hard" does not produce a finite result.
> time ToddCoxeter(Rs, sub<Rs|> : Strategy := "Hard", Style := "C"); 0 Time: 22015.849The best strategy is, to simplify the presentation obtained from the Reidemeister-Schreier procedure by eliminating generators until the total length of the relators starts to grow.
> time Rsl := SimplifyLength(R); Time: 0.330 > NumberOfGenerators(Rsl); 48 > PresentationLength(Rsl); 7152A coset enumeration for this presentation produces a finite result in a reasonable amount of time.
> time ToddCoxeter(Rsl, sub<Rsl|> : Strategy := "Hard", Style := "C"); 32928 Time: 289.410This finally proves that G is finite and has order 32928 * 114 = 3753792.