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The maximal subgroups of the finite 8-dimensional orthogonal groups \(P\Omega ^ +_ 8(q)\) and of their automorphism groups. (English) Zbl 0623.20031
Let p be a prime number, \(q=p^ n\). Let \(G_ 0=P\Omega^+_ 8(q)\cong D_ 4(q)\) be the simple Chevalley group of type \(D_ 4\) over the field of q elements and \(G_ 0\trianglelefteq G\leq Aut G\). The author gives a complete list of maximal subgroups of G. The classification theorem of finite simple groups is used in the proof. Note that the similar problem for other classical groups of dimension \(\leq 12\) is solved in the author’s article [”The low-dimensional finite classical groups and their subgroups” (to appear)].
Reviewer: A.Zalesskij

MSC:
20G40 Linear algebraic groups over finite fields
20D06 Simple groups: alternating groups and groups of Lie type
20D45 Automorphisms of abstract finite groups
20H30 Other matrix groups over finite fields
20D30 Series and lattices of subgroups
20E28 Maximal subgroups
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References:
[1] Artin, E, ()
[2] Aschbacher, M, On the maximal subgroups of the finite classical groups, Invent. math., 76, 469-514, (1984) · Zbl 0537.20023
[3] {\scM. Aschbacher}, Chevalley groups of type G2 as the groups of a trilinear form, preprint. · Zbl 0618.20030
[4] Borel, A; Tits, J, Elements unipotents et sous groupes paraboliques de groupes reductifs, Invent. math., 12, 95-104, (1971) · Zbl 0238.20055
[5] Brauer, R; Nesbitt, C.J, On the modular characters of groups, Ann. math., 42, 556-590, (1941), (2) · JFM 67.0073.02
[6] Butler, G; McKay, J, The transitive groups of degree up to eleven, Comm. algebra, 11, 8, 863-911, (1983) · Zbl 0518.20003
[7] Carter, R.W, Simple groups of Lie type, (1972), Wiley London · Zbl 0248.20015
[8] Chastkofsky, L; Feit, W, On the projective characters in characteristic 2 of the groups suz(2m) and sp4(2n), Inst. hautes études sci., 51, 9-36, (1980)
[9] Conway, J.H; Curtis, R.T; Norton, S.P; Parker, R.A; Wilson, R.A, ()
[10] Cooperstein, B.N, Maximal subgroups of G2(2n), J. algebra, 70, 23-36, (1981) · Zbl 0459.20007
[11] Dixon, J.D; Puttaswamaiah, B.M, Modular representations of finite groups, (1977), Academic Press New York · Zbl 0391.20004
[12] Dye, R.H, The simple group FH(8,2) of order 212.35.52.7 and the associated geometry of triality, (), 521-562 · Zbl 0159.31104
[13] Dye, R.H, Some geometry of triality with applications to involutions of certain orthogonal groups, (), 217-234 · Zbl 0215.11005
[14] Gorenstein, D; Lyons, R, The local structure of finite groups of characteristic 2 type, Mem. amer. math. soc., 42, 276, (1983) · Zbl 0519.20014
[15] Hartley, R.W, Determination of the ternary linear collineation groups whose coefficients Lie in the GF(2n), Ann. math., 27, 140-158, (1926) · JFM 51.0117.07
[16] Humphreys, J.F, The projective characters of the Mathieu group M12 and of its automorphism group, (), 401-412 · Zbl 0429.20015
[17] Humphreys, J.F, The projective characters of the Mathieu group M22, J. algebra, 76, 1-24, (1982) · Zbl 0489.20010
[18] James, G.D, The modular characters of the Mathieu groups, J. algebra, 27, 57-111, (1973) · Zbl 0268.20008
[19] Janko, Z, A new finite simple group with abelian Sylow 2-subgroups and its characterization, J. algebra, 3, 147-186, (1966) · Zbl 0214.28003
[20] {\scW. M. Kantor}, “Classical Groups from a Non-Classical Viewpoint,” Mathematical Institute, Oxford. · Zbl 0505.20034
[21] Khosraviyani, F, Decomposition numbers of exceptional Weyl groups, I, J. algebra, 91, 390-409, (1984) · Zbl 0559.20011
[22] Khosraviyani, F; Morris, A.O, Decomposition numbers of exceptional Weyl groups, II, J. algebra, 92, 525-531, (1985) · Zbl 0656.20018
[23] {\scP. B. Kleidman}, The maximal subgroups of the Steinberg triality groups ^{3}D4(q) and of their automorphism groups, J. Algebra, to appear. · Zbl 0642.20013
[24] {\scP. B. Kleidman}, The low-dimensional finite simple classical groups and their subgroups, in preparation. · Zbl 0623.20031
[25] {\scP. B. Kleidman}, The maximal subgroups of the Chevalley groups G2(q) with q odd, the Ree groups 2G2(q), and of their automorphism groups, J. Algebra, to appear. · Zbl 0651.20020
[26] Landazuri, V; Seitz, G.M, On the minimal degrees of projective representations of the finite Chevalley groups, J. algebra, 32, 418-443, (1974) · Zbl 0325.20008
[27] Liebeck, M.W, On the orders of maximal subgroups of the finite classical groups, (), 426-446, (3) · Zbl 0591.20021
[28] {\scM. W. Liebeck}, The affine permutation groups of rank three, Proc. London Math. Soc., to appear. · Zbl 0621.20001
[29] {\scM. W. Liebeck, C. E. Praeger, and J. Saxl}, A classification of the maximal subgroups of the finite alternating and symmetric groups, J. Algebra, to appear. · Zbl 0632.20011
[30] Miglione, E.T, The determination of the maximal subgroups of G2(q), q odd, ()
[31] Mitchell, H.H, Determination of the ordinary and modular ternary linear groups, Trans. amer. math. soc., 12, 207-242, (1911) · JFM 42.0161.01
[32] Mitchell, H.H, The subgroups of the quaternary abelian linear group, Trans. amer. math. soc., 15, 377-396, (1914) · JFM 45.0252.03
[33] Moore, E, The subgroups of the generalized modular group, (), 141-190
[34] {\scR. A. Parker}, “Modular Character Tables,” University of Cambridge, Cambridge.
[35] {\scJ. Sarli}, Thesis, U.C.S.C.
[36] Steinberg, R, Endomorphisms of linear algebraic groups, Mem. amer. math. soc., 80, (1968) · Zbl 0164.02902
[37] Suzuki, M, On a class of doubly transitive groups, Ann. math., 75, 105-145, (1962) · Zbl 0106.24702
[38] Wagner, A, The faithful linear representations of least degree of Sn and an over a field of characteristic 2, Math. Z., 151, 127-137, (1976) · Zbl 0321.20008
[39] Wagner, A, The faithful linear representations of least degree of Sn and an over a field of odd characteristic, Math. Z., 154, 103-114, (1977) · Zbl 0336.20008
[40] Wagner, A, An observation on the degrees of projective representations of the symmetric and alternating group over an arbitrary field, Arch. math., 29, 583-589, (1977) · Zbl 0383.20009
[41] Wilson, R.A, Maximal subgroups of automorphism groups of simple groups, J. London math. soc., 97, 460-466, (1985) · Zbl 0562.20006
[42] Wiman, A, Bestimmung aller untergruppen einer doppelt unendlichen reihe von endlichen gruppen, Bihang. till K. svenska vet. akad. handlingar, 25, 1, 1-47, (1899) · JFM 30.0197.01
[43] Zalesskii, A.E, Linear groups, Russian math. surveys, 36, 5, 63-128, (1981) · Zbl 0501.20024
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