On a two-fold cover 2.(2⁶˙G₂(2)) of a maximal subgroup of Rudvalis group Ru

• Prins, Abraham Love ^[1]
  1. [1] Nelson Mandela University.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 40, Nº. 4, 2021, págs. 1011-1029
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-4574
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• Resumen

  • The Schur multiplier M(Ḡ1) ≅4 of the maximal subgroup Ḡ1 = 2⁶˙G₂(2)of the Rudvalis sporadic simple group Ru is a cyclic group of order 4. Hence a full representative group R of the type R = 4.(2⁶˙G₂(2)) exists for Ḡ1. Furthermore, Ḡ1 will have four sets IrrProj(Ḡ1;αi) of irreducible projective characters, where the associated factor sets α1, α2, α3 and α4, have orders of 1, 2, 4 and 4, respectively. In this paper, we will deal with a 2-fold cover 2. Ḡ1 of Ḡ1 which can be treated as a non-split extension of the form Ḡ = 27˙G2(2). The ordinary character table of Ḡ will be computed using the technique of the so-called Fischer matrices. Routines written in the computer algebra system GAP will be presented to compute the conjugacy classes and Fischer matrices of Ḡ and as well as the sizes of the sets |IrrProj(Hi; αi)| associated with each inertia factor Hi. From the ordinary irreducible characters Irr(Ḡ) of Ḡ, the set IrrProj(Ḡ1; α2) of irreducible projective characters of Ḡ1 with factor set α2 such that α22= 1, can be obtained.

• Referencias bibliográficas
  • F. Ali and J. Moori, “The Fischer-Clifford matrices of a maximal subgroup of Fi’24”, Representation theory, vol. 7, pp. 300-321, 2003. https://doi.org/10.1090/S1088-4165-03-00175-4
  • A. B. M. Basheer and J. Moori, “A survey on Clifford-Fischer theory”, in Groups St Andrews 2013, C. M. Campbell, M. R. Quick, E. F. Robertson,...
  • W. Bosma and J. J. Cannon, Handbook of Magma Functions. Sidney: University of Sydney, 1995.
  • C. Chileshe, “Irreducible characters of Sylow p-Subgroups associated with some classical linear groups”, Ph.D. Thesis, Mathematics, North-West...
  • J. H. Conway, R.T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups. Oxford: Clarendon Press, 1985.
  • B. Fischer, “Clifford-matrices”, in Representation theory of finite groups and finite-dimensional algebras, G. O. Michler and C. Ringel, Eds....
  • The GAP Group, “GAP - Groups, Algorithms, Programming - a system for computational discrete algebra”, Version 4.11.0. Feb-2020 [Online]. Available:...
  • D. Gorenstein, Finite groups. New York, NY: Harper and Row, 1968.
  • R. J. Haggarty and J. F. Humphreys, “Projective characters of finite groups”, Proceedings of the London Mathematical Society, vol. s3-36,...
  • I. M. Isaacs, Character theory of finite groups. San Diego, CA: Academic Press, 1976.
  • G. Karpilovsky, Group representations, vol. 1-B. Amsterdam: North Holland, 1992.
  • G. Karpilovsky, Projective representations of finite groups. New York, NY: Marcel Dekker, 1985.
  • J. Moori, “On certain groups associated with the smallest Fischer group”, Journal of the London Mathematical Society, vol. s2-23, no. 1, pp....
  • J. Moori and Z. E. Mpono, “The Fischer-Clifford matrices of the group 26:SP6(2)”, Quaestiones mathematicae, vol. 22, no. 2, pp. 257-298, 1999....
  • A. L. Prins and R. L. Fray, “The Fischer-Clifford matrices of the inertia group 27:O−6(2) of a maximal subgroup 27:Sp6(2) in Sp8(2)”, International...
  • A. L. Prins, “On the Fischer-Clifford matrices of the non-split extension 26· G2(2)”, Bulletin of the Iranian Mathematical Society, vol. 41,...
  • A. L. Prins, “The projective character tables of a solvable group 26:(6 × 2)”, International journal of mathematics and mathematical sciences,...
  • A. L. Prins, “A maximal subgroup 24+6:(A5 × 3) of G2(4) treated as a non-split extension Ḡ = 26· (24:(A5 × 3))”, Advances in group...
  • A. L. Prins, “On the projective character tables of the maximal subgroups of M11, M12 and Aut(M12)”, Advances in group theory and applications,...
  • A. L. Prins, R. L. Monaledi and R. L. Fray, “On a maximal subgroup (29:L3(4)):3 of the automorphism group U6(2):3 of U6(2)”, Afrika matematika,...
  • T. T. Seretlo, “Fischer Clifford Matrices and Character Tables of Certain Groups Associated with Simple Groups O+10(2), HS and Ly”, PhD...
  • G. Robinson, “The number of irreducible projective characters with associated factor set of any finite group”, MathOverflow, 05-May-2014 [Online]....
  • J. Schmidt, “Projective characters with corresponding factor set”, MathOverflow, 13-Apr-2017 [Online]. Available: https://bit.ly/3y4Liwv
  • R. A. Wilson, P. Walsh, J. Tripp, I. Suleiman, S. Rogers, R. Parker, S. Norton, S. Nickerson, S. Linton, J. Bray and R. Abbot, ATLAS of Finite...