Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T04:54:44.341Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  31 October 2024

Robert T. Curtis
Robert T. Curtis
Affiliation:
University of Birmingham
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
The Art of Working with the Mathieu Group M24 , pp. 276 - 280
Publisher: Cambridge University Press
Print publication year: 2024

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Assmus, E. F., and Key, J. D. (1992), Designs and Their Codes. Cambridge University Press.CrossRefGoogle Scholar
Baker, H. F. (1935), ‘Note introductory to Klein’s group of order 168’, Proc. Cambridge Phil. Soc., 31, 468–481.CrossRefGoogle Scholar
Bannai, E., and Ito, T. (1984), Algebraic Combinatorics I Association Schemes. Benjamin/Cummings Publishing Company.Google Scholar
Benson, D. J. (1980), The Simple Group J4, PhD. thesis, Cambridge University Press.Google Scholar
Berlekamp, E. R., Conway, J. H., and Guy, R. K. (1982), Winning Ways for Your Mathematical Plays, Vol. 2. Academic Press.Google Scholar
Bolt, S. (2002), Some Applications of Symmetric Generation, PhD. thesis, University of Birmingham.Google Scholar
Borcherds, R. E. (1998), What Is Moonshine? Documenta Mathematica, ICM, 607–615.Google Scholar
Borcherds, R. E., Conway, J. H., Queen, L., and Sloane, N. J. A. (1984), ‘A Monster Lie algebra’, Adv. Math., 53 (1), 75–79.CrossRefGoogle Scholar
Bosma, W., and Cannon, J. (1994), Handbook of MAGMA Functions. University of Sydney.Google Scholar
Bray, J. N. (1997), Symmetric Presentations of Finite Groups, PhD. thesis, University of Birmingham.Google Scholar
Bray, J. N., and Curtis, R. T. (2003), ‘Monomial modular representations and symmetric generation of the Harada–Norton group’, J. Algebra, 268, 723–743.CrossRefGoogle Scholar
Bray, J. N. and Curtis, R. T. (2004), ‘Double coset enumeration of symmetrically generated groups’, J. Group Theory., 7, 167–185.CrossRefGoogle Scholar
Bray, J. N., Holt, D. F., and Roney-Dougal, C. M. (2013), The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. London Mathematical Society Lecture Note Series, Vol. 407, Cambridge University Press.Google Scholar
Burness, T. and Giudici, M. (2016), Classical Groups, Derangements and Primes. Australian Mathematical Society Lecture Series, Vol. 25, Cambridge University Press.Google Scholar
Cameron, P. J. (1994), Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press.Google Scholar
Cameron, P. J. (1999), Permutation Groups. LMS Student texts, Vol. 45, Cambridge University Press.Google Scholar
Cameron, P. J., and van Lint, J. H. (1991), Graphs, Codes, Designs and Their Links. Cambridge University Press.CrossRefGoogle Scholar
Carter, R. W. 1972, reprinted (1989), Simple Groups of Lie Type. Wiley.Google Scholar
Chevalley, C. (1955), ‘Sur certains groupes simples’, Tˆohoku Math. J., 7, 14–66.Google Scholar
Choi, C. (1972), ‘On subgroups of M24. II: the maximal subgroups of M24’, Trans. Am. Math. Soc., 167, 29–47.Google Scholar
Cohn, H., Kumar, A., Miller, S. D., Radchenko, D., and Viazovska, M. (2022), ‘Universal optimality of the E8 and Leech lattices and interpolation formulas’, Ann. Math., 196(3), 983–1082.CrossRefGoogle Scholar
Collins, M. J. (1990), Representations and Characters of Finite Groups. Cambridge Studies in Advanced Mathematics, Vol. 22, Cambridge University Press.Google Scholar
Conway, J. H. (1969a), ‘A characterisation of Leech’s lattice’, Inventiones Math., 7, 137–142.CrossRefGoogle Scholar
Conway, J. H. (1969b), ‘A group of order 8,315,553,613,086,720,000’, Bull. Lond. Math. Soc., 1, 79–88.CrossRefGoogle Scholar
Conway, J. H. (1971), ‘Three lectures on exceptional groups’. Pages 215–247 of: Powell, M. B., and Higman, G. (eds), Finite Simple Groups. Academic Press.Google Scholar
Conway, J. H. (1973), ‘A construction of the smallest Fischer group’. Pages 27–35 of: Gagen, T., Hale, M. P., and Shult, E. E. (eds.), Finite Groups ’72. North Holland.Google Scholar
Conway, J. H. (1981), ‘The hunting of J4’, Eureka, 41, 46–54.Google Scholar
Conway, J. H. (1983), ‘The automorphism group of the 26-dimensional even unimodular Lorentz lattice’, J. Algebra., 80, 159–163.CrossRefGoogle Scholar
Conway, J. H. (1985), ‘A simple construction of the Fischer-Griess Monster group’, Invent. Math., 79 (3), 513–540.CrossRefGoogle Scholar
Conway, J. H., and Norton, S. P. (1979), ‘Monstrous Moonshine’, BLMS, 11, 308–339.Google Scholar
Conway, J. H., and Sloane, Neil A. J. (1988), Sphere-Packing, Lattices and Groups. Springer-Verlag.CrossRefGoogle Scholar
Conway, J. H., Norton, S. P. and Soicher, L.H. (1985), ‘The Bimonster, the group Y555, and the projective plane of order 3’. Pages 27–50 of: Tangora, M.C. (ed.), Computers in Algebra (Chicago, Il. 1985), Lecture Notes in Pure and Applied Mathematics, Vol. 111. Marcel Dekker (1988).Google Scholar
Conway, J. H., Parker, R. A., and Sloane, N. J. A. (1982), ‘The covering radius of the Leech lattice’, Proc. Roy. Soc., 380(1779), 261–290.Google Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., and Wilson, R. A. (1985), An Atlas of Finite Groups. Clarendon Press.Google Scholar
Curtis, R. T. (1972), The Mathieu Group M24 and Related Topics. PhD thesis, University of Cambridge.Google Scholar
Curtis, R. T. (1976), ‘A new combinatorial approach to M24’, Math. Proc. Cambridge Phil. Soc., 79, 25–42.CrossRefGoogle Scholar
Curtis, R. T. (1977), ‘The maximal subgroups of M24’, Math. Proc. Cambridge Phil. Soc., 81, 185–192.CrossRefGoogle Scholar
Curtis, R. T. (1984), ‘Eight octads suffice’, J. Comb. Theory., 36, 116–123.CrossRefGoogle Scholar
Curtis, R. T. (1989), ‘Natural constructions of the Mathieu groups’, Math. Proc. Cambridge Phil. Soc., 106, 423–429.CrossRefGoogle Scholar
Curtis, R. T. (1990), ‘Geometric interpretations of the “natural” constructions of the Mathieu groups’, Math. Proc. Cambridge Phil. Soc., 107, 19–26.CrossRefGoogle Scholar
Curtis, R. T. (1993), ‘Symmetric generation II: The Janko group J1’, J. London Math. Soc., 47, 294–308.Google Scholar
Curtis, R. T. (1996), ‘Monomial modular representations and construction of the Held group’, J. Algebra, 184, 1205–1227.CrossRefGoogle Scholar
Curtis, R. T. (2007a), ‘Construction of a family of Moufang loops’, Math. Proc. Cambridge Phil. Soc., 142(2), 233–237.CrossRefGoogle Scholar
Curtis, R. T. (2007b), Symmetric Generation of Groups. Encyclopedia of Mathematics and Its Applications, Vol. 111, Cambridge University Press.Google Scholar
Curtis, R. T. (2016), ‘The Thompson chain of subgroups of the Conway group Co1 and complete graphs on n vertices’, J. Group Theory., 19(6), 959–982.CrossRefGoogle Scholar
Curtis, R. T. (2017), ‘Construction of the Thompson Chain of subgroups of the Conway group O and complete graphs on n letters’. Pages 73–90 of: Bhargava, M., Guralnick, R., Hiss, G., Lux, K., and Tiep, P. H. (eds), Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemporary Mathematics, Vol. 694. American Mathematical Society.Google Scholar
Curtis, C. W., and Reiner, I. (1962), Representation Theory of Finite Groups and Associative Algebras. John Wiley & Sons.Google Scholar
Dickson, L. E. (1919), ‘On quaternions and their generalisation and the history of the eight squares theorem’, Annal. Math., 20, 155–171.CrossRefGoogle Scholar
Dickson, L. E. (2007), Linear Groups, with an Exposition of the Galois Field Theory. Cosimo Classics.Google Scholar
Dieudonné, J. (1963), La Géometrie des Groupes Classiques. 2nd edn. Springer-Verlag.CrossRefGoogle Scholar
Graves, J. T. (1844), On the sum of eight squares theorem and the inderlying system of imaginaries. In letters to William Rowan Hamilton.Google Scholar
Griess, R. L. (1998), Twelve Sporadic Groups. Springer.CrossRefGoogle Scholar
Halberstam, H., and Ingram, R. E. (eds). (1967), The Mathematical Papers of Sir William Rowan Hamilton, Vol. III, Algebra. Cambridge University Press.Google Scholar
Hall, M., and Wales, D. B. (1969), ‘A simple group of order 604,800’. Pages 79–90 of: Brauer, R., and Sah, C.-H. (eds), The Theory of Finite Groups. W. A. Benjamin.Google Scholar
Harada, K. (1976), On the simple group F of order 214.36.56.7.11.19. In: Proceedings of the Conference on Finite Groups (Utah 1975). Academic Press.Google Scholar
Held, D. (1969), ‘Some simple groups related to M24’. Pages 121–124 of: Brauer, R., and Sah, C.-H. (eds), The Theory of Finite Groups. W. A. Benjamin.Google Scholar
Hilton, H. (1920), Plane Algebraic Curves. Oxford University Press.Google Scholar
Hurwitz, A. (1923), ‘Über die Komposition der quadratischen Formen’, Math. Ann. 88, 1–25.CrossRefGoogle Scholar
Isaacs, I. M. (1976), Character Theory of Finite Groups. Pure and Applied Mathematics. Academic Press.Google Scholar
Ivanov, A. A. (2018), The Mathieu Group M24. Cambridge University Press.Google Scholar
Jacobson, N. (1980), Basic Algebra II. W. H. Freeman and Company.Google Scholar
Jacobson, N. (1985), Basic Algebra I. 2nd edn. W. H. Freeman and Company.Google Scholar
James, G., and Liebeck, M. (1993), Representations and Characters of Groups. Cambridge University Press.Google Scholar
Janko, Z. (1965), ‘A new finite simple group with abelian Sylow 2-subgroups’, Proc. Natl. Acad. Sci. USA, 53, 675–658.CrossRefGoogle Scholar
Janko, Z. (1966), ‘A new finite simple group with abelian Sylow 2-subgroups and its characterization’, J. Algebra, 3, 147–186.CrossRefGoogle Scholar
Janko, Z. (1969), Some new simple groups of finite order. Pages 63–64 of: Brauer, R., and Sah, C.-H. (eds), The Theory of Finite Groups. W. A. Benjamin.Google Scholar
Janko, Z. (1976), ‘A new finite simple group of order 86, 775, 570, 046, 077, 562, 880, which possesses M24 and the full covering group of M22 as subgroups’, J. Algebra, 42, 564–596.CrossRefGoogle Scholar
Jónsson, W. (1972), ‘On the Mathieu groups M22, M23, M24 and the uniqueness of the associated Steiner systems’, Math Z., 125, 193–214.CrossRefGoogle Scholar
King, O. H. (2005), The Subgroup Structure of Finite Classical Groups in Terms of Geometric Configurations. London Mathematical Society Lecture Note Series, Vol. 327. Cambridge University Press.Google Scholar
Kleidman, P. B., and Liebeck, M. W. (1990), The Subgroup Structure of the Finite Classical Groups. Cambridge University Press.CrossRefGoogle Scholar
Klein, F. (1878), ‘Uber die transformation siebenter ordnung der elliptischen functionen’, Gesammelte Math. Abhandlungen, 90–135.Google Scholar
Leavitt, D. W., and Magliveras, S. S. (1984), Pages 337–352 of: Simple 6 − (33, 6, 36) designs from PΓL2 (32), Computational Group Theory. Academic Press.Google Scholar
Leech, J. (1956), ‘The problem of the 13 spheres’, Math. Gazette., 40, 22–23.CrossRefGoogle Scholar
Leech, J. (1964), ‘Some sphere packings in higher space’, Canad. J. Math., 16, 657–682.CrossRefGoogle Scholar
Lempken, W. (1978), ‘A 2-local characterization of the Janko’s simple group J4’, J. Algebra, 55, 403–445.CrossRefGoogle Scholar
Lyons, R. (1972), ‘Evidence for a new finite simple group’, J. Algebra, 20, 540–569.CrossRefGoogle Scholar
MacWilliams, F. J., and Sloane, N. J. A. (1977), The Theory of Error-Correcting Codes. North Holland.Google Scholar
Mathieu, E. (1861), ‘Memoire sur l’étude des fonctions de plusieurs quantités’, J. Math. Pure Appl., 6, 241–243.Google Scholar
Mathieu, E. (1873), ‘Sur les fonctions cinq fois transitive de 24 quantités’, J. Math. Pure Appl., 18, 25–46.Google Scholar
Mordell, L. J. (1969), Diophantine Equations. Academic Press.Google Scholar
Neubueser, J., et al. (2006), Groups, Algorithms and Programming. Aachen.Google Scholar
Neumaier, A., and Seidel, J. J. (1983), ‘Discrete hyperbolic geometry’, Combinatorica, 3, 219–237.CrossRefGoogle Scholar
Niemeier, H.-V. (1973), ‘Definite quadratische Formen der Dimension 24 und Diskrim-inante 1’, J. Number Theory, 5, 142–178.CrossRefGoogle Scholar
Norton, S. P. (1975), F and Other Simple Groups, PhD thesis, Cambridge University Press.Google Scholar
Norton, S. P. (1980), ‘The construction of J4’, Proc. of Symposia in Pure Math., 37, 271–278.CrossRefGoogle Scholar
O’Nan, M. E. (1976), ‘Some evidence for the existence of a new simple group’, Proc. London Math. Soc., 32, 421–479.CrossRefGoogle Scholar
Parker, R. A., Wilson, R. A., Bray, J. N. et al. (1999) Atlas of Finite Group Representations. http://brauer.maths.qmul.ac.uk/Atlas/Google Scholar
Praeger, C. E., and Soicher, L. H. (1997), Low Rank Representations and Graphs for Sporadic Groups. Australian Mathematical Society Lecture Series B. Cambridge University Press, xi+141.Google Scholar
Ree, R. (1960), ‘A family of simple groups associated with the simple Lie algebra (G2)’, Bulletin of the American Math. Soc., 66, 508–510.CrossRefGoogle Scholar
Ree, R. (1961), ‘A family of simple groups associated with the simple Lie algebra of type (F4)’, Bulletin of the American Math. Soc., 67, 115–116.CrossRefGoogle Scholar
Reid, M. A. (2023), Sextactic Points. Unpublished response to a question from the author.Google Scholar
Rice, A. (2004), John Thomas Graves (1806–1870). Oxford Dictionary of National Biography. Oxford University Press.Google Scholar
Rowley, P. (2005), ‘A Monster graph’, Proc. London Mathematical Society (3), 90(1), 42–60.CrossRefGoogle Scholar
Rowley, Peter, and Walker, L. (2004a), ‘A 11,707,448,673,375 vertex graph related to the Baby Monster I’, J. Combinatorial Theory, 107(2), 181–213.CrossRefGoogle Scholar
Rowley, P., and Walker, L. (2004b), ‘A 11,707,448,673,375 vertex graph related to the Baby Monster II’, J. Combinatorial Theory, 107(2), 215–261.CrossRefGoogle Scholar
Rowley, P., and Walker, L. (2012), ‘A 195,747,435 vertex graph related to the Fischer group Fi23, III’, JP Journal of Algebra, Number Theory and Appl., 27, 1–44.Google Scholar
Rowley, P., and Walker, L. (2021), ‘The point-line collinearity graph of the Fi′24 maximal 2-local geometry - the first three discs’, JANTAA, 24(1), 53–109.Google Scholar
Rudvalis, A. (1973), ‘A new simple group of order 214.33.53.7.13.29’, Notices American Math. Soc., 20, A–95.Google Scholar
Suzuki, M. (1964), ‘On a class of doubly transitive groups’, Ann. Math., 79, 514–589.CrossRefGoogle Scholar
Taormina, A., and Wendland, K. (2013), ‘The overarching symmetry group of Kummer surfaces in the Mathieu group M24’, J. High Energy Physics, 125(8), 1–63.Google Scholar
Taormina, A., and Wendland, K. (2015a), ‘Symmetry-surfing the moduli space of Kummer K3s’, Proc. Symp. Pure Math., 90, 129–154.CrossRefGoogle Scholar
Taormina, A., and Wendland, K. (2015b), ‘A twist in the M24 moonshine theory’, Confluentes Mathematici, 7, 83–113.CrossRefGoogle Scholar
Tits, J. (1980a), ‘Four presentations of Leech’s lattice’. Pages 303–307 of: Collins, M.J. (ed), Finite Simple Groups II. Academic Press.Google Scholar
Tits, J. (1980b), ‘Quaternions over ℚ(√5), Leech’s lattice and the sporadic group of Hall-Janko’, J. Algebra, 63, 56–75.CrossRefGoogle Scholar
Todd, J. A. (1966), ‘Representations of the Mathieu group M24 as a collineation group’, Ann. di Math. Pure ed Appl. 71, 199–238.CrossRefGoogle Scholar
Todd, J. A. (1970), ‘Abstract definitions for the Mathieu groups’, Quart. J. Math. Oxford, 21, 421–424.CrossRefGoogle Scholar
Weyl, H. (1953), The Classical Groups, Their Invariants and Representations. 2nd edn. Princeton Landmarks in Mathematics. Princeton University Press.Google Scholar
Wilson, R. A. (1982), ‘The quaternionic lattice for 2G2(4) and its maximal subgroups’, J. Algebra, 77, 449–466.CrossRefGoogle Scholar
Wilson, R. A. (1983), ‘The complex Leech lattice and maximal subgroups of the Suzuki group’, J. Algebra, 84, 151–188.CrossRefGoogle Scholar
Wilson, R. A. (1986), ‘Is J1 a subgroup of the Monster?’ Bull. London Math. Soc., 18, 349–350.CrossRefGoogle Scholar
Wilson, R. A. (2009a), The Finite Simple Groups. Graduate Texts in Mathematics, Vol. 251, Springer-Verlag.Google Scholar
Wilson, R. A. (2009b), ‘Octonions and the Leech lattice’, J. Algebra, 322, 2186–2190.CrossRefGoogle Scholar
Witt, E. (1938a). ‘Die 5-fach transitiven Gruppen von Mathieu’, Abh. Math. Sem. Hamb., 12, 256–265.CrossRefGoogle Scholar
Witt, E. (1938b). ‘Uber Steinersche Systeme’, Abh. Math. Sem. Hamb., 12, 265–275.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Robert T. Curtis, University of Birmingham
  • Book: The Art of Working with the Mathieu Group M24
  • Online publication: 31 October 2024
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Robert T. Curtis, University of Birmingham
  • Book: The Art of Working with the Mathieu Group M24
  • Online publication: 31 October 2024
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Robert T. Curtis, University of Birmingham
  • Book: The Art of Working with the Mathieu Group M24
  • Online publication: 31 October 2024
Available formats
×