×

zbMATH — the first resource for mathematics

The factorizations of the finite exceptional groups of Lie type. (English) Zbl 0607.20012
In this paper the factorizations \(G=AB\) of exceptional finite simple groups G of Lie type are determined. It is shown, that only \(G_ 2(q)\) and \(F_ 4(q)\) have factorizations and these factorizations are given explicitly. By a former result of two of the authors [Preprint, Univ. Cambridge (1985)] one knows the structure of A if one assumes \(| A| \geq | B|\). Secondly, if G is defined over GF(q), one looks for primitive divisors of \(q^ a-1\) where a is ”big” and observes that such primes occur in \(| B|\). These facts then can be played off against each other effectively to obtain the result.
Reviewer: U.Dempwolff

MSC:
20D40 Products of subgroups of abstract finite groups
20D06 Simple groups: alternating groups and groups of Lie type
20G40 Linear algebraic groups over finite fields
PDF BibTeX Cite
Full Text: DOI
References:
[1] Borel, A; Tits, J, Élements unipotents et sous-groupes paraboliques de groupes réductifs, Invent. math., 12, 469-514, (1971) · Zbl 0238.20055
[2] Carter, R.W, Conjugacy classes in the Weyl group, () · Zbl 0254.17005
[3] Chang, B, The conjugate classes of Chevalley groups of type (G2), J. algebra, 9, 190-211, (1968) · Zbl 0285.20043
[4] Conway, J.H; Curtis, R.T; Norton, S.P; Parker, R.A; Wilson, R.A, ()
[5] Deriziotis, D.I, The centralizers of semisimple elements of the Chevalley groups E7 and E8, Tokyo J. math., 6, 191-216, (1983) · Zbl 0534.20031
[6] Deriziotis, D.I; Liebeck, M.W, Centralizers of semisimple elements in finite twisted groups of Lie type, J. London math. soc., 31, 48-54, (1985) · Zbl 0599.20064
[7] Enomoto, H, The conjugacy classes of Chevalley groups of type (G2) over finite fields of characteristic 2 or 3, J. fac. sci. univ. Tokyo, 16, 497-512, (1970) · Zbl 0242.20049
[8] Fisman, E, On the product of two finite solvable groups, J. algebra, 80, 517-536, (1983) · Zbl 0503.20005
[9] Gorenstein, D; Lyons, R, The local structure of finite groups of characteristic 2 type, Mem. amer. math. soc., 42, (1983) · Zbl 0519.20014
[10] Hering, C, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geom. dedicata, 2, 425-460, (1974) · Zbl 0292.20045
[11] Ito, N, Über das produkt von zwei abelschen gruppen, Math. Z., 62, 400-401, (1955) · Zbl 0064.25203
[12] Kegel, O.H, Produkte nilpotenter gruppen, Arch. math., 12, 90-93, (1961) · Zbl 0099.01401
[13] 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
[14] Liebeck, M.W, On the orders of maximal subgroups of the finite classical groups, (), 426-446 · Zbl 0591.20021
[15] {\scM. W. Liebeck, C. E. Praeger, and J. Saxl}, The factorizations of the finite simple groups and their automorphism groups, in preparation. · Zbl 0703.20021
[16] {\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
[17] Liebeck, M.W; Saxl, J, Primitive permutation groups containing an element of large prime order, J. London math. soc., 31, 237-249, (1985) · Zbl 0573.20003
[18] {\scM. W. Liebeck and J. Saxl}, On the orders of maximal subgroups of the finite exceptional groups of Lie type, submitted. · Zbl 0627.20026
[19] Mizuno, K, The conjugacy classes of Chevalley groups of type E6, J. fac. sci. univ. Tokyo, 24, 525-563, (1977) · Zbl 0399.20044
[20] Preiser, U, Factorizations of finite groups, Math. Z., 185, 373-402, (1984) · Zbl 0536.20013
[21] Shinoda, K, The conjugacy classes of Chevalley groups of type F4 over finite fields of characteristic 2, J. fac. sci. univ. Tokyo, 21, 133-159, (1974) · Zbl 0306.20013
[22] Shinoda, K, The conjugacy classes of the finite ree groups of type F4, J. fac. sci. univ. Tokyo, 22, 1-15, (1975) · Zbl 0306.20014
[23] Shoji, T, The conjugacy classes of Chevalley groups of type F4 over finite fields of characteristic p ≠2, J. fac. sci. univ. Tokyo, 21, 1-17, (1974) · Zbl 0279.20038
[24] Springer, T.A; Steinberg, R, Conjugacy classes, () · Zbl 0249.20024
[25] Suzuki, M, On a class of doubly transitive groups, Ann. of math., 75, 105-145, (1962) · Zbl 0106.24702
[26] Szép, J, Sui gruppi fattorizzabili non semplici, Rend. mat. appl., 22, 245-252, (1963) · Zbl 0152.00501
[27] Tchakerian, K.B; Gentchev, T.R, Factorizations of the groups G2(q), Arch. math., 44, 230-232, (1985) · Zbl 0544.20012
[28] Ward, H.N, On Ree’s series of simple groups, Trans. amer. math. soc., 121, 62-89, (1966) · Zbl 0139.24902
[29] Wiegold, J; Williamson, A.G, The factorisation of the alternating and symmetric groups, Math. Z., 175, 171-179, (1980) · Zbl 0424.20004
[30] Zsigmondy, K, Zur theorie der potenzreste, Monatsh. math. phys., 3, 265-284, (1892) · JFM 24.0176.02
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.