56.4 ANU Sq Package

Sq( G, L )

The function Sq is the interface to the Soluble Quotient standalone program.

Let G be a finitely presented group and let L be a list of lists. Each of these lists is a list of integer pairs [p_i,c_i], where p_i is a prime and c_i is a non-negative integer and p_i not= p_{i+1} and c_i positive for i < k. Sq computes a consistent power conjugate presentation for a finite soluble group given as a quotient of the finitely presented group G which is described by L as follows.

Let H be a group and p a prime. The series H = P^p_0(H)ge P^p_1(H)gecdots,,hbox to 1.5 cmhfil with P^p_i(H) = [P^p_i-1(H),H] left(P^p_i-1 (H)right)^p for i ge 1 is the it lower exponent-p central series of H.

For 1 le i le k and 0 le j le c_i define the list {rm L}_{i,j} = [(p_1,c_1),ldots, (p_{i-1},c_{i-1}),(p_{i}, j) ]. Define {rm L}_{1,0}(G) = G. For 1 le i le k and 1 le j le c_i define the subgroups rm L_i,j(G) = rm P_j^p_i( rm L_i,0(G) ) and for 1 le i < k define the subgroups rm L_i+1,0(G) = rm L_i,c_i(G) and {rm L}(G) = {rm L}_{k,c_k}(G). Note that {rm L}_{i,j}(G) ge {rm L}_{i,j+1}(G) holds for j < c_i.

The chain of subgroups G = rm L_1,0(G) ge rm L_1,1(G) ge cdots ge rm L_1,c_1(G) = rm L_2,0(G)ge cdots ge rm L_k,c_k(G) = rm L(G) is called the it soluble {rm L}-series of G.

Sq computes a consistent power conjugate presentation for G/{rm L}(G), where the presentation exhibits a composition series of the quotient group which is a refinement of the soluble {rm L}-series. An epimorphism from G onto G/{rm L}(G) is listed in comments.

The algorithm proceeds by computing power conjugate presentations for the quotients G/{rm L}_{i,j}(G) in turn. Without loss of generality assume that a power conjugate presentation for G/{rm L}_{i,j}(G) has been computed for j < c_i. The basic step computes a power conjugate presentation for G/{rm L}_{i,j+1}(G). The group {rm L}_{i,j}(G)/{rm L}_{i,j+1}(G) is a p_{i}-group. If during the basic step it is discovered that {rm L}_{i,j}(G) = {rm L}_{i,j+1}(G), then {rm L}_{i+1,0}(G) is set to {rm L}_{i,j}(G).

Note that during the basic step the vector enumerator is called.

    gap> RequirePackage("anusq");
    gap> f := FreeGroup( "a", "b" );;
    gap> f := f/[ (f.1*f.2)^2*f.2^-6, f.1^4*f.2^-1*f.1*f.2^-9*f.1^-1*f.2 ];
    Group( a, b )
    gap> g := Sq( f, [[2,1],[3,1],[2,2],[3,2]] );
    rec(
      generators := [ a.1, a.2, a.3, a.4, a.5, a.6, a.7, a.8 ],
      relators :=
       [ a.1^2*a.3^-1, a.1^-1*a.2*a.1*a.4^-1*a.2^-2, a.2^3*a.5^-1,
          a.1^-1*a.3*a.1*a.3^-1,
          a.2^-1*a.3*a.2*a.6^-1*a.5^-1*a.4^-1*a.3^-1, a.3^2*a.7^-1*a.5^-1,
          a.1^-1*a.4*a.1*a.7^-1*a.4^-1*a.3^-1,
          a.2^-1*a.4*a.2*a.8^-1*a.7^-1*a.6^-2*a.3^-1,
          a.3^-1*a.4*a.3*a.8^-2*a.7^-2*a.5^-1*a.4^-1,
          a.4^2*a.8^-2*a.7^-2*a.6^-2*a.5^-1,
          a.1^-1*a.5*a.1*a.8^-1*a.7^-1*a.6^-1*a.5^-1,
          a.2^-1*a.5*a.2*a.5^-1, a.3^-1*a.5*a.3*a.8^-2*a.6^-1*a.5^-1,
          a.4^-1*a.5*a.4*a.7^-1*a.5^-1, a.5^2,
          a.1^-1*a.6*a.1*a.8^-1*a.7^-2*a.6^-1,
          a.2^-1*a.6*a.2*a.8^-2*a.6^-2,
          a.3^-1*a.6*a.3*a.8^-2*a.7^-2*a.6^-2,
          a.4^-1*a.6*a.4*a.8^-1*a.7^-2, a.5^-1*a.6*a.5*a.8^-2*a.6^-2,
          a.6^3, a.1^-1*a.7*a.1*a.6^-2, a.2^-1*a.7*a.2*a.7^-2*a.6^-2,
          a.3^-1*a.7*a.3*a.8^-1*a.7^-1*a.6^-2, a.4^-1*a.7*a.4*a.6^-1,
          a.5^-1*a.7*a.5*a.7^-2, a.6^-1*a.7*a.6*a.8^-1*a.7^-1, a.7^3,
          a.1^-1*a.8*a.1*a.8^-2, a.2^-1*a.8*a.2*a.8^-1,
          a.3^-1*a.8*a.3*a.8^-1, a.4^-1*a.8*a.4*a.8^-1,
          a.5^-1*a.8*a.5*a.8^-1, a.6^-1*a.8*a.6*a.8^-1,
          a.7^-1*a.8*a.7*a.8^-1, a.8^3 ] ) 

This implementation was developed in C by

Alice C. Niemeyer
Department of Mathematics
University of Western Australia
Nedlands, WA 6009
Australia

email:

alice@maths.uwa.edu.au

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