Afrika Matematika

, Volume 28, Issue 7–8, pp 1011–1032 | Cite as

On two groups of the form \(2^{8}{:}A_{9}\)

Article
First Online:
Received:
Accepted:
  • 64 Downloads

Abstract

This paper is dealing with two split extensions of the form \(2^{8}{:}A_{9}.\) We refer to these two groups by \(\overline{G}_{1}\) and \(\overline{G}_{2}.\) For \(\overline{G}_{1},\) the 8-dimensional GF(2)-module is in fact the deleted permutation module for \(A_{9}.\) We firstly determine the conjugacy classes of \(\overline{G}_{1}\) and \(\overline{G}_{2}\) using the coset analysis technique. The structures of inertia factor groups were determined for the two extensions. The inertia factor groups of \(\overline{G}_{1}\) are \(A_{9},\,A_{8},\, S_{7},\,(A_{6} \times 3){:}2 \) and \((A_{5} \times A_{4}){:}2,\) while the inertia factor groups of \(\overline{G}_{2}\) are \(A_{9},\, PSL(2,8){:}3\) and \(2^{3}{:}GL(3,2).\) We then determine the Fischer matrices for these two groups and apply the Clifford–Fischer theory to compute the ordinary character tables of \(\overline{G}_{1}\) and \(\overline{G}_{2}.\) The Fischer matrices of \(\overline{G}_{1}\) and \(\overline{G}_{2}\) are all integer valued, with sizes ranging from 1 to 9 and from 1 to 4 respectively. The full character tables of \(\overline{G}_{1}\) and \(\overline{G}_{2}\) are \(84 \times 84\) and \(40 \times 40\) complex valued matrices respectively.

Keywords

Group extensions Alternating group Inertia groups Fischer matrices Character table 

Mathematics Subject Classification

20C15 20C40 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The authors would like to thank the referees for the corrections and valuable suggestions and comments, which improved the manuscript. The first author would like to thank his supervisor (second author) for his advice and support. Also the North-West University (Mafikeng) and the National Research Foundation (NRF) of South Africa are acknowledged for their financial supports.

References

  1. 1.
    Basheer, A.B.M.: Clifford–Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups, Ph.D. Thesis. University of KwaZulu-Natal, Pietermaitzburg (2012)Google Scholar
  2. 2.
    Basheer, A.B.M., Moori, J.: Fischer matrices of Dempwolff group \(2^{5}{^{\cdot }}GL(5,2)\). Int. J. Group Theory 1(4), 43–63 (2012)MATHMathSciNetGoogle Scholar
  3. 3.
    Basheer, A.B.M., Moori, J.: On the non-split extension group \(2^{6}{^{\cdot }}Sp(6,2)\). Bull. Iran. Math. Soc. 39(6), 1189–1212 (2013)MATHMathSciNetGoogle Scholar
  4. 4.
    Basheer, A.B.M., Moori, J.: On the non-split extension \(2^{2n}{^{\cdot }}Sp(2n,2)\). Bull. Iran. Math. Soc. 41(2), 499–518 (2015)MATHMathSciNetGoogle Scholar
  5. 5.
    Basheer, A.B.M., Moori, J.: On a maximal subgroup of the Thompson simple group. Math. Commun. 20, 201–218 (2015)MATHMathSciNetGoogle Scholar
  6. 6.
    Basheer, A.B.M., Moori, J.: A survey on Clifford–Fischer theory. London Mathematical Society Lecture Notes Series, vol. 422, pp. 160–172. Cambridge University Press, Cambridge (2015)Google Scholar
  7. 7.
    Basheer, A.B.M., Moori, J.: On a group of the form \(3^{7}{:}Sp(6,2)\). Int. J. Group Theory 5(2), 41–59 (2016)MATHMathSciNetGoogle Scholar
  8. 8.
    Basheer, A.B.M., Moori, J.: On the deleted permutation modules of the alternating group \(A_{n}\) (in preparation) Google Scholar
  9. 9.
    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)MATHGoogle Scholar
  10. 10.
    The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.4.10 (2007). http://www.gap-system.org
  11. 11.
    Isaacs, M.: Character Theory of Finite Groups. Academic Press, New York (1976)MATHGoogle Scholar
  12. 12.
    Jansen, C., Lux, K., Parker, R., Wilson, R.: An Atlas of Brauer Characters, London Mathematical Society Monographs, New Series. Clarenden Press, Oxford (1995)MATHGoogle Scholar
  13. 13.
    Maxima: A Computer Algebra System. Version 5.18.1 (2009). http://maxima.sourceforge.net
  14. 14.
    Moori, J.: On the Groups \(G^{+}\) and \(\overline{G}\) of the form \(2^{10}{:}M_{22}\) and \(2^{10}{:}\overline{M}_{22}\), Ph.D. Thesis. University of Birmingham (1975)Google Scholar
  15. 15.
    Moori, J.: On certain groups associated with the smallest Fischer group. J. Lond. Math. Soc. 2, 61–67 (1981)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Moori, J., Seretlo, T.T.: On two non-split extension groups associated with \(HS\) and \(HS{:}2\). Turk. J. Math. 37, 60–78 (2013)MATHGoogle Scholar
  17. 17.
    Moori, J., Seretlo, T.T.: On the Fischer–Clifford matrices of a maximal subgroup of the Lyons Group \(Ly\). Bull. Iran. Math. Soc. 39, 1037–1052 (2013)MATHMathSciNetGoogle Scholar
  18. 18.
    Mpono, Z.E.: Fischer Clifford Theory and Character Tables of Group Extensions, Ph.D. Thesis. University of Natal, Pietermaritzburg (1998)Google Scholar
  19. 19.
    Seretlo, T.T.: Fischer Clifford Matrices and Character Tables of Certain Groups Associated with Simple Groups \(O_{10}^+(2),~HS\) and \(Ly\), Ph.D. thesis. University of KwaZulu-Natal, Pietermaritzburg (2012)Google Scholar
  20. 20.
    Whitely, N.S.: Fischer Matrices and Character Tables of Group Extensions, M.Sc. Thesis. University of Natal, Pietermaritzburg (1993)Google Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorth-West University (Mafikeng)MmabathoSouth Africa

Personalised recommendations