Refined solvable presentations for polycyclic groups

Document Type: Research Paper

Authors

Abstract

‎We describe a new type of presentation that‎, ‎when consistent‎, ‎describes a polycyclic group‎. ‎This presentation is obtained by‎ ‎refining a series of normal subgroups with‎ ‎abelian sections‎. ‎These presentations can be described effectively in‎ ‎computer-algebra-systems like $ Gap$ or‎ ‎$ Magma$‎. ‎We study these‎ ‎presentations and‎, ‎in particular‎, ‎we obtain consistency criteria for them‎. ‎The‎ ‎consistency implementation demonstrates that there are situations where‎ ‎the new method is faster than the‎ ‎existing methods for polycyclic groups‎.

Keywords

Main Subjects


B. Assmann and S. Linton (2007). Using the Mal'cev correspondence for collection in polycyclic groups. J. Algebra. 316 (2), 828-848
G. Baumslag and D. Solitar (1962). Some two-generator one-relator non-Hopfian groups. Bull. Amer. Math. Soc.. 68, 199-201
L. Bartholdi (2003). Endomorphic presentations of branch groups. J. Algebra. 268 (2), 419-443
L. Bartholdi, B. Eick and R. Hartung (2008). A nilpotent quotient algorithm for certain infinitely presented groups and its applications. Internat. J. Algebra Comput.. 18 (8), 1321-1344
A. M. Brunner, S. Sidki and A. C. Vieira (1999). A just nonsolvable torsion-free group defined on the binary tree. J. Algebra. 211 (1), 99-114
R. I. Grigorchuk and A. Zuk (2002). On a torsion-free weakly branch group defined by a three state automaton. Internat. J. Algebra Comput.. 12 (1-2), 223-246
G. Endimioni and G. Traustason (2008). Groups that are pairwise nilpotent. Comm. Algebra. 36 (12), 4413-4435
The GAP Group. (2008). GAP-- Groups, Algorithms, and Programming. 4.4.12
V. Gebhardt (2002). Efficient collection in infinite polycyclic groups. J. Symbolic Comput.. 34 (3), 213-228
R. Hartung (2009). NQL- Nilpotent quotients of L-presented groups. An accepted Gap 4 package,.
R. Hartung (2010). Approximating the Schur multiplier of certain infinitely presented groups via nilpotent quotients. LMS J. Comput. Math.. 13, 260-271
C. R. Leedham-Green and L. H. Soicher (1990). Collection from the left and other strategies. J. Symbolic Comput.. 9 (5-6), 665-675
E. H. Lo (1997). A polycyclic quotient algorithm. In Groups and computation, II (New Brunswick, NJ, 1995) volume 28 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 159--167. Amer. Math. Soc., Providence, RI, 1997.. 28, 159-167
E. H. Lo (1998). A polycyclic quotient algorithm. J. Symbolic Comput.. 25 (1), 61-97
W. Nickel (1996). Computing nilpotent quotients of finitely presented groups. In Geometric and computational perspectives on infinite groups (Minneapolis, NN and New Brunswick, NJ, 1994), volume 25 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 175--191. Amer. Math. Soc., Providence, RI, 1996.. 25, 175-191
C. C. Sims (1994). Computation with finitely presented groups. Cambridge University Press, New York.
M. R. Vaughan-Lee (1984). An aspect of the nilpotent quotient algorithm. Computational Group Theory (Durham 1982); Academic Press, London. , 75-83
M. R. Vaughan-Lee (1990). Collection from the left. J. Symbolic Comput.. 9 (5-6), 725-733