Abstract
The treatment of quasithin groups of characteristic 2 was one of the last steps in the Classification of the finite simple groups. Geoff Mason [12] announced a classification of these groups in about 1980, but never published his work. A few people have a copy of a large manuscript containing his efforts, but because it was distributed slowly, section by section, it was only during the last few years that it was realized that Mason’s manuscript is incomplete in various ways. A few years ago I wrote up a treatment which begins where Mason’s manuscript ends and finishes the problem assuming the results he says he proves. I have only read Mason’s manuscript superficially, but it appears there are missing lemmas even for the part of the problem the theorems in his manuscript cover. I do believe however that he has seriously addressed the issues involved and that he could turn his manuscript into a proof with enough work. However Mason is now involved with Moonshine and has no interest in completing or publishing his manuscript.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aschbacher, M. (1978) Thin finite simple groups, J. Algebra 54, 50–152.
Aschbacher, M. (1986) Overgroups of Sylow subgroups in sporadic groups, Mern. Amer. Math. Soc. 343, 1–235.
Aschbacher, M. (1993) Simple connectivity of p-group complexes, Israel J. Math. 82, 1–42.
Delgado, A., Goldschmidt, D. and Stellmacher, B. (1987) Groups and Graphs: New Results and Methods, Birkhäuser, Basel.
Fong, P. and Seitz, G. (1973) Groups with a (B,N)-pair of rank 2, I, Invent. Math. 21, 1–57.
Fong, P. and Seitz, G. (1974) Groups with a (B,N)-pair of rank 2, II, Invent. Math. 24, 191–239.
Goldschmidt, D. (1980) Automorphisms of trivalent graphs, Ann. Math. 111, 377–406.
Gomi, K. and Hayashi, M. (1992) A pushing up approach to the quasithin finite simple groups with solvable 2-local subgroups, J. Algebra 146, 412–426.
Glauberman, G. and Niles, R. (1983) A pair of characteristic subgroups for pushing up in finite groups, Proc. London Math. Soc. 46, 411–453.
Janko, Z. (1972) Nonsolvable finite groups all of whose 2-local subgroups are solvable, I, J. Algebra 21, 458–517.
Janko, Z. and Wong, S.K. (1969) A characterization of the Higman-Sims simple group, J. Algebra 13, 517–534.
Mason, G. (n.d.) The classification of finite quasithin groups, preprint.
Meierfrankenfeld, U. and Stellmacher, B. (1993) Pushing up weak BN-pairs of rank 2, Comm. in Alg. 21, 825–934.
Parrott, D. and Wong, S.K. (1970) On the Higman-Sims simple group of order 44, 352, 000, Pac. J. Math. 32 501–516.
Serre, J.-P. (1980) Trees, Springer-Verlag, Berlin.
Sims, C. (1967) Graphs and finite permutation groups, Math. Zeit. 95 76–86.
Smith, F. (1975) Finite simple groups all of whose 2-local subgroups are solvable, J. Algebra 34 481–520.
Stellmacher, B. (1992) On the 2-local structure of finite groups, in M. Liebeck and J. Saxl (eds.), Groups, Combinatorics, and Geometry, Cambridge University Press, Cambridge, pp. 159–182.
Stellmacher, B. (1997) An application of the amalgam method: The 2-local structure of N-groups of characteristic 2 type, to appear.
Suzuki, M. (1965) Finite groups in which the centralizer of any element of order 2 is 2-closed, Ann. Math. 82, 191–212.
Thompson, J. (1968) Nonsolvable groups all of whose local subgroups are solvable, I, Bull. Amer. Math. Soc. 74, 383–437.
Thompson, J. (1970) Nonsolvable groups all of whose local subgroups are solvable, II, Pac. J. Math. 33 451–536.
Thompson, J. (1971) Nonsolvable groups all of whose local subgroups are solvable, III, Pac. J. Math. 39 483–534.
Thompson, J. (1973) Nonsolvable groups all of whose local subgroups are solvable, IV, Pac. J. Math. 48, 511–592.
Thompson, J. (1974) Nonsolvable groups all of whose local subgroups are solvable, V, Pac. J. Math. 50 215–297.
Thompson, J. (1974) Nonsolvable groups all of whose local subgroups are solvable, VI, Pac. J. Math. 51 573–630.
Tutte, W. (1959) On the symmetry of cubic graphs, Canadian J. Math. 11, 621–624.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Aschbacher, M. (1998). Quasithin Groups. In: Carter, R.W., Saxl, J. (eds) Algebraic Groups and their Representations. NATO ASI Series, vol 517. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5308-9_18
Download citation
DOI: https://doi.org/10.1007/978-94-011-5308-9_18
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-7923-5292-1
Online ISBN: 978-94-011-5308-9
eBook Packages: Springer Book Archive