25.35 PQuotient

PQuotient( G, p, cl )
PrimeQuotient( G, p, cl )

PQuotient computes quotients of prime power order of finitely presented groups. G must be a group given by generators and relations. PQuotient expects G to be a record with the record fields generators and relators. The record field generators must be a list of abstract generators created by the function AbstractGenerator (see AbstractGenerator). The record field relators must be a list of words in the generators which are the relators of the group. p must be a prime. cl has to be an integer, which specifies that the quotient of prime power order computed by PQuotient is the largest p-quotient of G of class at most cl. PQuotient returns a record Q, the PQp record, which has, among others, the following record fields describing the p-quotient Q.

generators:

A list of abstract generators which generate Q.

pcp :

The internal power-commutator presentation for Q.

dimensions:

A list, where dimensions[i] is the dimension of the i-th factor in the lower exponent-p central series calculated by the p-quotient algorithm.

prime:

The integer p, which is a prime.

definedby:

A list which contains the definition of the k-th generator in the k-th place. There are three different types of entries, namely lists, positive and negative integers.
[ j, i ]:
the generator is defined to be the commutator of the j-th and the i-th element in generators.
i:
the generator is defined as the p-th power of the i-th element in generators.
-i:
the generator is defined as an image of the i-th generator in the finite presentation for G, consequently it must be a generator of weight 1.

epimorphism:

A list containing an image in Q of each generator of G. The image is either an integer i if it is the i-th element of generators of Q or an abstract word w if it is the abstract word w in the generators of Q.

An example of the computation of the largest quotient of class 4 of the group given by the finite presentation { x,y mid x^{25}/(xcdot y)^5, [x,y]^5, (x^y)^{25} } .

    # Define the group
    gap> x := AbstractGenerator("x");;
    gap> y := AbstractGenerator("y");;
    gap> G := rec( generators := [x,y],
    >              relators := [ x^25/(x*y)^5, Comm(x,y)^5, (x^y)^25] );
    rec(
      generators := [ x, y ],
      relators :=
       [ x^25*y^-1*x^-1*y^-1*x^-1*y^-1*x^-1*y^-1*x^-1*y^-1*x^-1,
          x^-1*y^-1*x*y*x^-1*y^-1*x*y*x^-1*y^-1*x*y*x^-1*y^-1*x*y*x^-1*y^-\
    1*x*y, y^-1*x^25*y ] )

    # Call pQuotient
    gap> P := PQuotient( G, 5, 4 );
    #I  PQuotient: class 1 : 2
    #I  PQuotient: Runtime : 0
    #I  PQuotient: class 2 : 2
    #I  PQuotient: Runtime : 27
    #I  PQuotient: class 3 : 2
    #I  PQuotient: Runtime : 1437
    #I  PQuotient: class 4 : 3
    #I  PQuotient: Runtime : 1515
    PQp( rec(
       generators  := [ g1, g2, a3, a4, a6, a7, a11, a12, a14 ],
       definedby   := [ -1, -2, [ 2, 1 ], 1, [ 3, 1 ], [ 3, 2 ],
      [ 5, 1 ], [ 5, 2 ], [ 6, 2 ] ],
       prime       := 5,
       dimensions  := [ 2, 2, 2, 3 ],
       epimorphism := [ 1, 2 ],
       powerRelators := [ g1^5/(a4), g2^5/(a4^4), a3^5, a4^5, a6^5, a7^
    5, a11^5, a12^5, a14^5 ],
       commutatorRelators := [ Comm(g2,g1)/(a3), Comm(a3,g1)/(a6), Comm(a3\
    ,g2)/(a7), Comm(a6,g1)/(a11), Comm(a6,g2)/(a12), Comm(a7,g1)/(a12), Co\
    mm(a7,g2)/(a14) ],
       definingCommutators := [ [ 2, 1 ], [ 3, 1 ], [ 3, 2 ], [ 5, 1 ],
      [ 5, 2 ], [ 6, 1 ], [ 6, 2 ] ] ) )

The p-quotient algorithm returns a PQp record for the exponent-5 class 4 quotient. Note that instead of printing the PQp record P an equivalent representation is printed which can be read in to GAP. See PQp for details.

The quotient defined by P has nine generators, g1, g2, a3, a4, a6, a7,a11, a12, a14, stored in the list P.generators. From powerRelators we can read off that g1^5 =: a4 and g2^5 = a4^4 and all other generators have trivial 5-th powers. From the list commutatorRelators we can read off the non-trivial commutator relations Comm(g2,g1) =: a3, Comm(a3,g1) =: a6, Comm(a3,g2) =: a7, Comm(a6,g1) =: a11,Comm(a6,g2) =: a12, Comm(a7,g1) = a12 and Comm(a7,g2) =: a14. In this list =: denotes that the generator on the right hand side is defined as the left hand side. This information is given by the list definedby. The list dimensions shows that P is a class-4 quotient of order 5^2cdot 5^2cdot 5^2cdot 5^3 = 5^9. The epimorphism of G onto the quotient P is given by the map x mapsto g1 and y mapsto g2.

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