×

Sporadic groups, code loops and nonvanishing cohomology. (English) Zbl 0611.20009

The aim of this article is to introduce a unified description of a certain exceptional behavior of the degree 1- and 2-cohomology of certain groups of Lie type. This work was initiated by the construction of a particular 2-local subgroup of the Fischer-Griess monster group by the automorphisms of a certain Moufang loop [J. H. Conway, Invent. Math. 79, 513-540 (1985; Zbl 0564.20010)]. Generalizing this concept the author defines a special class of Moufang loops - code loops - which are extensions of elementary abelian 2-groups V. He shows that \(H^ 2(G,V)\neq 0\), where G lies in the automorphism group of the code loop satisfying a specific condition on involutions (for this condition see also the author’s work [in Pac. J. Math. 48, 403-422 (1973; Zbl 0283.20028)]). In more detail a code loop of order 16, a subloop of the nonzero Cayley numbers, with the nonsplit extension \(2^ 3GL(3,2)\) as automorphisms is studied. Other extensions discussed with the ”loop method” by the author are \(4^ 3GL(3,2)\) (Higman-Sims and O’Nan group), and \(2^{3+8}GL(3,2)\) (Rudvalis group).
Reviewer: U.Dempwolff

MSC:

20D08 Simple groups: sporadic groups
20J06 Cohomology of groups
20N05 Loops, quasigroups

References:

[1] Alperin, J., Sylow 2-subgroups of 2-rank 3, (Finite Groups ’72 Proc. Gainesville Conf., Univ. Florida. Finite Groups ’72 Proc. Gainesville Conf., Univ. Florida, Gainesville, FL, 1972. Finite Groups ’72 Proc. Gainesville Conf., Univ. Florida. Finite Groups ’72 Proc. Gainesville Conf., Univ. Florida, Gainesville, FL, 1972, North-Holland Mathematical Studies, Vol. 7 (1973), North-Holland: North-Holland Amsterdam), 1-12 · Zbl 0269.20011
[2] Avrunin, G., The image of the restriction map on 2-cohomology, Arch. Math. (Basel), 34, 502-508 (1980) · Zbl 0436.20032
[3] Benson, D., Modular Representation Theory via Representation Rings, (Lecture Notes in Math. (1985), Springer: Springer Berlin) · Zbl 0648.20014
[4] Blackburn, N., The extension theory of the symmetric and alternating groups, Math. Z., 117, 191-206 (1970) · Zbl 0205.32401
[5] Cline, E.; Parshall, B.; Scott, L. L.; van der Kallen, W., Rational and generic cohomology, Invent. Math., 39, 143-163 (1977) · Zbl 0336.20036
[6] Cline, E.; Parshall, B.; Scott, L. L., Cohomology of finite groups of Lie type, I, Inst. Hautes Etuds Sci. Publ. Math., 45, 169-191 (1975) · Zbl 0412.20044
[7] Conway, J.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups (1985), Oxford University Press · Zbl 0568.20001
[8] J. Conway, A simple construction for the Fischer-Griess Monster group, Invent. Math.; J. Conway, A simple construction for the Fischer-Griess Monster group, Invent. Math. · Zbl 0564.20010
[9] Conway, J., Three lectures on exceptional groups, (Higman, G.; Powell, M., Finite Simple Groups (1969-1971), Academic Press: Academic Press London), Oxford
[10] Coxeter, H. S.M., Integral Cayley numbers, Duke Math. J., 13, 561-578 (1946) · Zbl 0063.01004
[11] Dempwolff, U., On the extensions of an elementary group of order \(2^5\) by GL(5, 2), Rend. Sem. Mat. Padova, 48, 359-364 (1972) · Zbl 0275.20017
[12] Dempwolff, U., On the second cohomology of GL \((n, 2)\), J. Austral. Math. Soc., 16, 207-209 (1973) · Zbl 0271.20024
[13] Dornhoff, L., Group Representation Theory (1971, 1972), Marcel Dekker: Marcel Dekker New York, 2 volumes · Zbl 0227.20002
[14] Feit, W., The Representation Theory of Finite Groups (1982), North-Holland: North-Holland Amsterdam · Zbl 0493.20007
[15] Friedlander, E. M., Homological stability for classical groups over finite fields, Algebraic \(K\)-Theory, (Proc. of a conference at Northwestern University in Evanston, IL. Proc. of a conference at Northwestern University in Evanston, IL, 1976. Proc. of a conference at Northwestern University in Evanston, IL. Proc. of a conference at Northwestern University in Evanston, IL, 1976, Lecture Notes in Math., 551 (1976), Springer: Springer Berlin), 290-302 · Zbl 0358.18013
[16] Friedlander, E. M.; Parschall, B., On the cohomology of Chevalley groups, Bull. Amer. Math. Soc., 7, 247-250 (1982) · Zbl 0492.20025
[17] Gagola, S.; Garrison, S., Real characters, double covers and the multiplier, J. Algebra, 74, 20-51 (1982) · Zbl 0479.20005
[18] Griess, R. L., Lecture at the Santa Cruz conference (1979)
[19] Griess, R. L., A sufficient condition for a finite group of even order to have nontrivial Schur multiplicator, (Notices Amer. Math. Soc. (1970))
[20] Griess, R. L., Automorphisms of extraspecial groups and nonvanishing degree 2 cohomology, Pacific J. Math., 48, 403-422 (1973) · Zbl 0283.20028
[21] Griess, R. L., Schur multipliers of the known finite simple groups, III, (Proc. of the Rutgers Group Theory Year 1983-1984 (1984), Cambridge University Press: Cambridge University Press Cambridge), 69-80 · Zbl 0648.20017
[22] R.L. Griess, Jr., Code loops, J. Algebra, to appear.; R.L. Griess, Jr., Code loops, J. Algebra, to appear. · Zbl 0589.20051
[23] Griess, R. L., The friendly giant, Invent. Math., 69, 1-102 (1982) · Zbl 0498.20013
[24] R.L. Griess, Jr., The monster and its nonassociative algebra, Proc. Montreal Conference on Finite Groups, to appear.; R.L. Griess, Jr., The monster and its nonassociative algebra, Proc. Montreal Conference on Finite Groups, to appear. · Zbl 0582.20007
[25] Higman, D. G., Flag transitive collineation groups of finite projective spaces, Illinois J. Math., 6, 434-446 (1962) · Zbl 0105.13101
[26] Huppert, B., Endliche Gruppen I (1967), Springer: Springer Berlin · Zbl 0217.07201
[27] Landazuri, V., Thesis (1975), University of Michigan
[28] O’Nan, M., Some evidence for the existence of a new simple group, (Proc. London Math. Soc., 32 (1976)), 421-479 · Zbl 0356.20020
[29] Ronan, M.; Smith, S. D., 2-local geometries for finite groups, (Proc. The Santa Cruz Conference on Finite Groups (1980), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · Zbl 0478.20015
[30] Schur, I., Ueber die Darstellung der endliche Gruppen durch gebrochene lineare Substitutionen, Crelle J. Math., 127, 20-50 (1904) · JFM 35.0155.01
[31] Schur, I., Untersuchungen ueber die Darstellungen der endlichen Gruppen durch gebrochene lineare Substitutionen, Crelle J. Math., 132, 85-137 (1907) · JFM 38.0174.02
[32] Schur, I., Ueber die Darstellungen der symmetrischen und alternierenden Gruppen durch gebrochene lineare Substitutionen, Crelle J. Math., 139, 155-250 (1911) · JFM 42.0154.02
[33] Steinberg, R., Générateurs, relations et revêtements de groupes algébriques, (Colloq. Théorie des Groupes Algébriques. Colloq. Théorie des Groupes Algébriques, Bruxelles, 1962 (1962), Gauthier-Villars: Gauthier-Villars Paris), 113-127, Librairie Universitaire, Louvain · Zbl 0272.20036
[34] Steinberg, R., Lectures on Chevalley groups, (Yale Lecture Notes (1967)) · Zbl 1361.20003
[35] Tits, J., Remarks on Griess’ construction of the Griess-Fischer sporadic group I, II, III, IV (1982-1983), Preprints distributed
[36] Tits, J., Théorie des groupes (1982-1983), Annuaire du Collège de France
[37] Tits, J., Le monstre, Séminaire Bourbaki, 620 (November 1983)
[38] Peterfalvi, Le théorème de Bender-Suzuki, I, Preprint.; Peterfalvi, Le théorème de Bender-Suzuki, I, Preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.