Subsections

• Introduction
• Rewriting

Let H be a subgroup of finite index in the finitely presented group G. It frequently happens that it is desirable to construct a set of defining relations for H from those of G. In principle, such a presentation can be obtained either on a set of Schreier generators for H using the Reidemeister-Schreier rewriting technique [W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Second Revised Edition, Dover, 1976.] or on the given generators of H [G. Arrell and E.F. Robertson, A modified Todd-Coxeter algorithm, in: M.D. Atkinson (ed) Computational Group Theory, Academic Press, 1984, pp. 27--32; G. Havas, P.E. Kenne, J.S. Richardson and E.F. Robertson, A Tietze transformation program, in: M.D. Atkinson (ed) Computational Group Theory, Academic Press, 1984, pp. 69--73 (and the references therein)].

If the user wishes only to determine the structure of the derived quotient group of H, then the function AbelianQuotientInvariants, described above, should be used. In this case there is no need to first construct a presentation for H using the Rewrite function described below, since AbelianQuotientInvariants employs a special form of the Reidemeister-Schreier rewriting process which abelianizes each relator as soon as it is constructed. Thus AbelianQuotientInvariants may be applied to subgroups H of much larger index in G than may be the function Rewrite.

Rewriting

Given a finitely presented group G and a subgroup H having finite index in G, construct a presentation for H by rewriting the presentation given for G.

    Simplify: BoolElt                   Default: true

If this Boolean-valued parameter is given the value true, then the resulting presentation for H will be simplified (default). The function Rewrite returns a finitely presented group that is isomorphic to H. If simplification is requested (Simplify := true) then the simplification procedures are invoked (see next section). These procedures perform a sequence of Tietze transformations which typically result in a considerable simplification of the presentation produced by the rewriting process. Alternatively, the user can set Simplify := false and then perform the simplification directly if desired. (See next section). If simplification is not requested as part of Rewrite, a small amount of simplification is performed on the presentation before it is returned.

    Iterations: RngIntElt               Default: 10000

Perform at most n iterations (default: 10000) of the main elimination loop.

    EliminationLimit: RngIntElt         Default: 100

Eliminate at most n generators (default: 100) in each iteration of the main elimination loop.

    LengthLimit: RngIntElt              Default: Infinity

Do not perform eliminations that make the total length of the current relations become larger than n (default: no limit).

> G<x, y> := Group< x, y | x^2, y^3, (x*y)^12, (x*y)^6*(x*y^-1)^6 >;
> G;

Finitely presented group G on 2 generators
Relations
       x^2 = Id(G)
       y^3 = Id(G)
       (x * y)^12 = Id(G)
       x * y * x * y * x * y * x * y * x * y * x * y * x * y^-1 * x * 
          y^-1 * x * y^-1 * x * y^-1 * x * y^-1 * x * y^-1 = Id(G)

> K := sub< G | x, y*x*y^-1, y*x*y^-1*x*y^-1*x*y >;
> K;
Finitely presented group K on 3 generators
Generators as words in group G
       K.1 = x
       K.2 = y * x * y^-1
       K.3 = y * x * y^-1 * x * y^-1 * x * y
> Index(G, K);
3
> T := Rewrite(G, K);
> T;
Finitely presented group T on 3 generators
Generators as words in group G
    T.1 = x
    T.2 = y * x * y^-1
    T.3 = y^3
Relations
    T.1^2 = Id(T)
    T.2^2 = Id(T)
    T.3^2 = Id(T)
    (T.3 * T.2 * T.1 * T.3 * T.2)^2 = Id(T)
    (T.1 * T.3 * T.2 * T.1 * T.3)^2 = Id(T)
    (T.1 * T.2 * T.1 * T.3 * T.2)^2 = Id(T)
    (T.3 * T.2 * T.3 * T.1 * T.2)^2 = Id(T)
> AbelianQuotientInvariants(T);
[ 2, 2, 2 ]

> Q2 := pQuotient(T, 2, 30);
> FactoredOrder(Q2);
[ <2, 62> ]