7.9 Implementation and examples
7.9.1 Runtimes
The algorithm IsPolycyclic of Section 7.4 was implemented in GAP [41] as a part of the Polenta package [3]. Instead of computing minimal polynomials in step (6) of the algorithm we determine characteristic polynomials because this is more efficient and because the minimal polynomial of a rational matrix is integral if and only if the characteristic polynomial is integral.
Alternatively the method in [8], which is also implemented in GAP, could be used to test whether the induced action ofG toQlog(U) conjugates into GL(e,Z). Therefore this method could replace the steps (6) to (11) of our algorithm “IsPolycyclic”. We compared this variation with our method and did not notice any difference in the runtimes for our example groups.
A method for testing virtual polycyclicity has not yet been implemented. To handle this case it is necessary to compute short finite presentations of finite non-soluble matrix groups.
In Table 7.4 we display runtimes for some example matrix groups and we also summarise some of the properties of the considered groups. All example groups considered in Table 7.4 are soluble and not contained in GL(d,Z). The groups G1, G2 are unipotent,G3, G4 are almost crystallographic groups. The groupG5 was constructed using the Kronecker product of generators of an almost crystallographic group. The group G6 is a randomly generated
88 Chapter 7. Alternatives beyond the Tits alternative subgroup of the direct product of a unipotent and a free-abelian-by-finite group. The group G7 is the group Gfrom the beginning of Section 7.9. The groupsG8, G9 are randomly generated upper-block-triangular matrix groups. Every example groupGi is available in the package Polenta via the func-
tion “SolvableMatGroupExams(i)”. A group G≤ GL(d,Q) given by gener- ators can be tested to be polycyclic using “IsPolycyclicMatGroup(G)”. All computations where carried out in GAP Version 4.4.7. on a 3.2 gigahertz Pentium 4 processor and 90 MB of memory for GAP.
Group Degree No. gens Rank DimQlog(U) Runtime
G1 4 2 4 4 47 G2 5 2 6 6 109 G3 5 5 4 4 822 G4 5 5 4 4 568 G5 16 5 3 3 557 G6 20 11 7 4 373083 G7 2 2 - 1 150 G8 6 4 - 10 15966 G9 8 4 - 13 14379
Table 7.4: Testing polycyclicity: The columns display the degree d, the number of generators, the rank (or Hirsch length) and the dimension of
Qlog(U) for every of the considered examplesG1, . . . , G9. If no rank is given, then the example group is not polycyclic. The last column contains the time in milliseconds which is needed by the algorithm IsPolycyclic of Section 7.4.
Appendix A
Algebraic number theory
In this appendix we recall some basic facts from algebraic number theory. For more background we refer to [40].
Definition A.0.1. Let Fbe a subfield of C such that [F:Q] is finite. Then
F is called a number field or an algebraic extension of Q. Let θ ∈ C. By
Q(θ) we denote the smallest subfield of C which contains θ.
Lemma A.0.2. LetF be a number field. Then there exists an elementθ ∈C
such that F=Q(θ).
Definition A.0.3. A complex numberθ is said to be analgebraic integer, if the minimal polynomial of θ is in Z[X].
Definition A.0.4. LetF be a number field. The set of algebraic integers in
F is called the maximal order O of F. An element u ∈ O is called a unit, if
u−1 is also contained in O. The set of all units in O is called the unit group of O and is denoted by U(O). A ring R ⊆ O with 1∈ R that spans F over
Q is called an order of F.
Theorem A.0.5 (Dirichlet’s Units Theorem). Let F be a number field with maximal order O. Then the torsion subgroup T of U(O) is finite and gen- erated by one element ζ ∈ U(O) which is called the torsion unit of U(O). Further there exists fundamental unitsε1, . . . , εr ∈U(O)such that every unit
u∈U(O) can be written uniquely as
u=ζf0 ·εf1 1 ·ε
f2 2 · · ·εfrr
where fi ∈Z and 0≤f0 <|T|.
Definition A.0.6. Letpbe a prime. The ring ofp-adic integersZp is defined to be the inverse limit of the finite quotients Z/pnZ.
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