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An \(L_2\)-quotient algorithm for finitely presented groups. (English) Zbl 1253.20033
Summary: The paper develops algorithmic methods to enumerate all normal subgroups of a finitely presented group such that the factor groups are either isomorphic to \(\mathrm{PSL}(2,p^n)\) or to \(\mathrm{PGL}(2,p^n)\). The case of two generators is treated in detail. A range of examples starting with the free group on two generators and ending with groups having only finitely many normal subgroups of this type is discussed.

20F05 Generators, relations, and presentations of groups
20-04 Software, source code, etc. for problems pertaining to group theory
68W30 Symbolic computation and algebraic computation
JanetOre; Magma; Janet
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[1] Blinkov, Y.A.; Cid, C.F.; Gerdt, V.P.; Plesken, W.; Robertz, D., The MAPLE package “janet”: I. polynomial systems, (), 31-40, also available together with the package from WWW:
[2] Blinkov, Y.A.; Gerdt, V.P.; Yanovich, D.A., Construction of janet bases, II. polynomial bases, (), 249-263 · Zbl 1015.13013
[3] Bosma, W.; Cannon, J.J.; Playoust, C., The magma algebra system I: the user language, J. symbolic comput., 24, 235-265, (1997) · Zbl 0898.68039
[4] Cavicchioli, A.; O’Brien, E.A.; Spaggiari, F., On some questions about a family of cyclically presented groups, J. algebra, 320, 11, 4063-4072, (2008) · Zbl 1201.20027
[5] Hall, P., The Eulerian functions of a group, (), 7, 179-196, (1936), also in
[6] Holt, D.F.; Rees, S., Finding subgroups and quotients of finitely presented groups, (), 99-107, (English summary) · Zbl 0829.20002
[7] Holt, D.F.; Plesken, W., A cohomological criterion for a finitely presented group to be infinite, J. London math. soc. (2), 45, 469-480, (1992) · Zbl 0792.20039
[8] Huppert, B., Endliche gruppen 1, (1967), Springer
[9] Kitaoka, Y., Arithmetic of quadratic forms, (1993), Cambridge Univ. Press · Zbl 0785.11021
[10] M. Lange-Hegermann, Algorithmen zur lokalen Kommutativen Algebra und Primärzerlegung, Diploma thesis Aachen, 2008
[11] Plesken, W.; Robertz, D., Janet’s approach to presentations and resolutions for polynomials and linear pdes, Arch. math. (basel), 84, 1, 22-37, (2005) · Zbl 1091.13018
[12] Plesken, W.; Robertz, D., Representations, commutative algebra, and Hurwitz groups, J. algebra, 300, 1, 223-247, (2006) · Zbl 1166.20011
[13] Plesken, W.; Souvignier, B., Analysing finitely presented groups by constructing representations, J. symbolic comput., 24, 335-349, (1997) · Zbl 0886.20024
[14] Plesken, W., Counting solutions of polynomial systems via iterated fibrations, Arch. math., 92, 44-56, (2009) · Zbl 1180.14053
[15] D. Robertz, Noether normalization guided by monomial cone decompositions, J. Symb. Comp., in press, 19 pp · Zbl 1190.13026
[16] Stevenhagen, P.; Lenstra, H.W., Chebotarev and his density theorem, Math. intelligencer, 18, 2, 26-37, (1996) · Zbl 0885.11005
[17] Suzuki, M., Group theory I, (1982), Springer
[18] Zassenhaus, H., On the spinor norm, Arch. math., 13, 434-451, (1962) · Zbl 0118.01804
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