Simple groups and Sylow subgroups

Koichiro Harada

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 6 (2013) no. 4, pp. 441-450.

Résumé

Firstly a problem to characterize Sylow $2$-subgroups (of small order $\leqslant 2^{10}$) of finite simple groups is proposed. Next some (possibly necessery) reduction steps are discussde. The latter half of these notes is devoted to finite groups having extra-spacial $p$-groups of order $p^3$ as Sylow subgroups.

Détail

Keywords: Sylow subgroups.
Mots-clés : simple group

Bibliographie

[1] M. Aschbacher, S. D. Smith, The classification of quasithin groups, v. I, Mathematical Surveys and Monographs, 111, Amer. Math. Soc., 2004 | DOI | MR | Zbl

[2] M. Aschabcher, R. Lyons, S. D. Smith, R. Solomon, The classification of finite simple groups: Groups of characteristic 2 type, Mathematical Surveys and Monographs, 172, Amer. Math. Soc., 2011 | MR

[3] D. Gorenstein, R. Lyons, R. Solomon, The classification of finite simple groups: Numbers 1,2,3,4,5, and 6, Mathematical Surveys and Monographs, 40, no. 1, Amer. Math. Soc., 1994 | DOI | MR | Zbl

[4] B. Oliver, Reduced fusion systems over 2-groups of sectional rank at most four, Preprint, 2013 | MR

[5] H. U. Besche, B. Eick, E. A. O'Brien, “The groups of order at most 2000”, Electron. Res. Announc. Amer. Math. Soc., 7 (2001), 1–4 | DOI | MR | Zbl

[6] B. Eick, E. A. O'Brien, “Enumerating $p$-groups”, J. Austral. Math. Soc., 7 (1999), 192–205

[7] E. A. O'Brien, “The groups of order 256”, J. Algebra, 143 (1991), 219–235 | DOI | MR

[8] R. James, M. F. Newman, E. A. O'Brien, “The groups of order 128”, J. Algebra, 129 (1990), 136–158 | DOI | MR | Zbl

[9] M. Hall (Jr.), J. K. Senior, The Groups of order $2^6$, ($n\leqslant6$), Macmillan, New York, 1964 | MR | Zbl

[10] D. Gorenstein, K. Harada, “Finite groups whose Sylow 2-subgroups are the direct product of two dihedral groups”, Ann. Math., 95 (1972), 1–54 | DOI | MR | Zbl

[11] D. M. Goldschmidt, “Strongly closed 2-subgroups of finite simple groups”, Ann. Math., 102 (1975), 475–489 | DOI | MR | Zbl

[12] K. Harada, Mong Lung Lang, “Indecomposable Sylow 2-subgroups of simple groups”, Acta Appl. Math., 85 (2005), 161–194 | DOI | MR | Zbl

[13] G. Higman, “Enumerating $p$-groups. I”, Proc. London Math. Soc. (3), 10 (1960), 24–30 | DOI | MR | Zbl

[14] C. C. Sims, “Enumerating $p$-groups”, Proc. London Math. Soc., 15 (1966), 151–156 | MR

[15] R. Gilman, D. Gorenstein, “Finite groups with Sylow 2-subgroups of class two. I”, Trans. Amer. Math. Soc., 207 (1975), 1–101 | DOI | MR | Zbl

[16] K. Gomi, “Finite groups all of whose non-2-closed 2-local subgroups have Sylow 2-subgroups of class 2”, J. Algebra, 35 (1975), 214–223 | DOI | MR | Zbl

[17] G. Glauberman, “Central elements in core-free groups”, J. Algebra, 4 (1966), 403–420 | DOI | MR | Zbl

[18] D. M. Golschmidt, “2-fusion in finite groups”, Ann. Math., 99 (1974), 70–117 | DOI | MR

[19] Tomoyuki Yoshida, “Character-theoretic transfer”, J. Algebra, 52 (1978), 1–38 | DOI | MR | Zbl

[20] A. Ruiz, A. Viruel, “The classification of $p$-local finite groups over the extraspecial group of order $p^3$ and exponent $p$”, Math. Zeit., 248:1 (2004), 45–65 | DOI | MR | Zbl

[21] L. E. Dickson, Linear groups with an exposition of the Galois field theory, Dover Publications, Inc., New York, 1958 | MR | Zbl

[22] Ryo Narasaki, Katsuhiro Uno, “Isometries and extra special Sylow groups of order $p^3$”, Jour. Alg., 322:6 (2009), 2027–2068 | DOI | MR | Zbl

[23] M. Linckelmann, “Book Review”, M. Aschbacher, R. Kessar and B. Oliver: Fusion systems in algebra and topology, London Math. Soc. Leture Notes, 391, Cambridge Univ. Press., 2011, and David A. Craven: The theory of fusion systems an algebraic approach, Cambridge Studies in Advanced Mathematics, 131, Cambridge Univ. Press., 2011, Bull. Amer. Math. Soc., 50 (2013), 523–526 | DOI | MR | Zbl

[24] D. Gorenstein, K. Harada, Finite groups whose 2-subgroups are generated by at most 4 elements, Mem. Amer. Math. Soc., 147, 1974, 464 pp. | MR | Zbl