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Computing Hall subgroups of finite groups

Part of: Group theory and generalizations Other groups of matrices Permutation groups Representation theory of groups

Published online by Cambridge University Press:  01 August 2012

Bettina Eick  and
Alexander Hulpke
Bettina Eick
Affiliation:
Institut Computational Mathematics, TU Braunschweig, D-38106 Braunschweig, Germany (email: [email protected])
Alexander Hulpke
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA (email: [email protected])

Abstract

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We describe an effective algorithm to compute a set of representatives for the conjugacy classes of Hall subgroups of a finite permutation or matrix group. Our algorithm uses the general approach of the so-called ‘trivial Fitting model’.

MSC classification

Secondary: 20B40: Computational methods 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure 20H20: Other matrix groups over fields 20-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 205 - 218
Copyright
Copyright © London Mathematical Society 2012

References

[1]Babai, L. and Beals, R., ‘A polynomial-time theory of black box groups I’, Groups St Andrews 1997 in Bath, I, London Mathematical Society Lecture Note Series 260 (eds Campbell, C. M., Robertson, E. F., Ruskuc, N. and Smith, G. C.; Cambridge University Press, Cambridge, 1999) 30–64.CrossRefGoogle Scholar
[2]Cannon, J., Cox, B. and Holt, D., ‘Computing Sylow subgroups in permutation groups’, J. Symbolic Comput. 24 (1997) no. 3/4, 303–316.CrossRefGoogle Scholar
[3]Cannon, J., Cox, B. and Holt, D., ‘Computing the subgroup lattice of a permutation group’, J. Symbolic Comput. 31 (2001) no. 1/2, 149–161.CrossRefGoogle Scholar
[4]Cannon, J. J. and Holt, D. F., ‘Automorphism group computation and isomorphism testing in finite groups’, J. Symbolic Comput. 35 (2003) no. 3, 241–267.CrossRefGoogle Scholar
[5]Cannon, J., Holt, D., Slattery, M. and Steel, A., ‘Computing subgroups of bounded index in a finite group’, J. Symbolic Comput. 40 (2005) no. 2, 1013–1022.CrossRefGoogle Scholar
[6]Cannon, J. J., Eick, B. and Leedham-Green, C. R., ‘Special polycyclic generating sequences for finite soluble groups’, J. Symbolic Comput. 38 (2004) no. 5, 1445–1460.CrossRefGoogle Scholar
[7]Celler, F., Neubüser, J. and Wright, C. R. B., ‘Some remarks on the computation of complements and normalizers in soluble groups’, Acta Appl. Math. 21 (1990) 57–76.CrossRefGoogle Scholar
[8]Feit, W., ‘Some consequences of the classification of finite simple groups’, The Santa Cruz Conference on Finite Groups (University of California, Santa Cruz, 1979), Proceedings of Symposia in Pure Mathematics 37 (American Mathematical Society, Providence, RI, 1980) 175–181.Google Scholar
[9]Feit, W. and Thompson, J., ‘Solvability of groups of odd order’, Pacific J. Math. 13 (1963) 775–1029.CrossRefGoogle Scholar
[10] The GAP group, ‘GAP—groups, algorithms and programming, version 4.4’, 2004,http://www.gap-system.org.Google Scholar
[11]Gross, F., ‘Conjugacy of odd order Hall subgroups’, Bull. Lond. Math. Soc. 19 (1987) no. 4, 311–319.CrossRefGoogle Scholar
[12]Gross, F., ‘Hall subgroups of order not divisible by 3’, Rocky Mountain J. Math. 23 (1993) no. 2, 569–591.CrossRefGoogle Scholar
[13]Gross, F., ‘Conjugacy of odd order Hall subgroups of the classical linear groups’, Math. Z. 220 (1995) no. 3, 317–336.CrossRefGoogle Scholar
[14]Hall, P., ‘Theorems like Sylow’s’, Proc. London Math. Soc. 6 (1956) 286–304.CrossRefGoogle Scholar
[15]Holt, D. F., ‘Representing quotients of permutation groups’, Quart. J. Math. Oxford Ser. (2) 48 (1997) no. 191, 347–350.CrossRefGoogle Scholar
[16]Holt, D. F., Eick, B. and O’Brien, E. A., Handbook of computational group theory, Discrete Mathematics and Its Applications (Chapman & Hall/CRC, Boca Raton, FL, 2005).CrossRefGoogle Scholar
[17]Hulpke, A., ‘Conjugacy classes in finite permutation groups via homomorphic images’, Math. Comp. 69 (2000) no. 232, 1633–1651.CrossRefGoogle Scholar
[18]Hulpke, A., ‘Computing conjugacy classes of elements in matrix groups’, in preparation, 2012.CrossRefGoogle Scholar
[19]Huppert, B., Endliche Gruppen I, Grundlehren der mathematischen Wissenschaften 134 (Springer, Berlin, 1967).CrossRefGoogle Scholar
[20]Kantor, W., ‘Sylow’s theorem in polynomial time’, J. Comput. System Sci. 30 (1985) no. 3, 359–394.CrossRefGoogle Scholar
[21]Kantor, W., ‘Finding Sylow normalizers in polynomial time’, J. Algorithms 11 (1990) no. 4, 523–563.CrossRefGoogle Scholar
[22]Luks, E. M. and Seress, Á., ‘Computing the Fitting subgroup and solvable radical for small-base permutation groups in nearly linear time’, Proceedings of the 2nd DIMACS Workshop held at Rutgers University, New Brunswick, NJ, June 7–10, 1995, DIMACS: Series in Discrete Mathematics and Theoretical Computer Science 28 (eds Finkelstein, Larry and Kantor, William M.; American Mathematical Society, Providence, RI, 1997) 169–181.Google Scholar
[23]O’Brien, E. A., ‘Algorithms for matrix groups’, Groups St Andrews 2009 in Bath. Volume 2, London Mathematical Society Lecture Note Series 388 (eds Campbell, C. M., Quick, M. R., Robertson, E. F., Roney-Dougal, C. M., Smith, G. C. and Traustason, G.; Cambridge University Press, Cambridge, 2011) 297–323.CrossRefGoogle Scholar
[24]Robinson, , A course in the theory of groups, 2nd edn (Springer, New York, 1996).CrossRefGoogle Scholar
[25]Schmalz, B., ‘Verwendung von Untergruppenleitern zur Bestimmung von Doppelnebenklassen’, Bayreuth. Math. Schr. 31 (1990) 109–143.Google Scholar
[26]Stather, M., ‘Constructive Sylow theorems for the classical groups’, J. Algebra 316 (2007) no. 2, 536–559.CrossRefGoogle Scholar
[27]Vdovin, E. P. and Revin, D. O., ‘Hall subgroups of odd order in finite groups’, Algebra Logika 41 (2002) no. 1, 8–29.CrossRefGoogle Scholar
[28]Revin, D. O. and Vdovin, E. P., ‘On the number of classes of conjugate Hall subgroups in finite simple groups’, J. Algebra 324 (2010) no. 12, 3614–3652.CrossRefGoogle Scholar
[29]Revin, D. O. and Vdovin, E. P., ‘Generalizations of the Sylow theorem’, Groups St Andrews 2009 in Bath. Volume 2, London Mathematical Society Lecture Note Series 388 (eds Campbell, , Quick, C. M., Robertson, M. R., Roney-Dougal, E. F., Smith, C. M. and Traustason, G. C.; Cambridge University Press, Cambridge, 2011) 488–519.CrossRefGoogle Scholar
[30]Thompson, J. G., ‘Hall subgroups of the symmetric groups’, J. Combin. Theory 1 (1966) 271–279.CrossRefGoogle Scholar