Arithmetical conditions on element orders and group structure

Zhang, Ji Ping

doi:10.1090/S0002-9939-1995-1239809-1

Arithmetical conditions on element orders and group structure

Author: Ji Ping Zhang
Journal: Proc. Amer. Math. Soc. 123 (1995), 39-44
MSC: Primary 20D60
DOI: https://doi.org/10.1090/S0002-9939-1995-1239809-1
MathSciNet review: 1239809
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Abstract: General results are provided on bounding the number of different prime factors of the order of finite groups in terms of the number for the order of elements.

Rolf Brandl, Finite groups all of whose elements are of prime power order, Boll. Un. Mat. Ital. A (5) 18 (1981), no. 3, 491–493 (English, with Italian summary). MR 633687
Michel Broué and Gunter Malle, Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis, Math. Ann. 292 (1992), no. 2, 241–262 (French). MR 1149033, DOI https://doi.org/10.1007/BF01444619
Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
Pamela A. Ferguson, Lengths of conjugacy classes of finite solvable groups. II, J. Algebra 154 (1993), no. 1, 223–227. MR 1201921, DOI https://doi.org/10.1006/jabr.1993.1013
David Gluck, Primes dividing character degrees and character orbit sizes, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 45–46. MR 933396
P. Hall and Graham Higman, On the $p$-length of $p$-soluble groups and reduction theorems for Burnside’s problem, Proc. London Math. Soc. (3) 6 (1956), 1–42. MR 72872, DOI https://doi.org/10.1112/plms/s3-6.1.1
Graham Higman, Finite groups in which every element has prime power order, J. London Math. Soc. 32 (1957), 335–342. MR 89205, DOI https://doi.org/10.1112/jlms/s1-32.3.335
Bertram Huppert, Inequalities for character degrees of solvable groups, Arch. Math. (Basel) 46 (1986), no. 5, 387–392. MR 847081, DOI https://doi.org/10.1007/BF01210777
Hua Lo-keng, Su lung tao yeng, Science Press, Peking, 1964 (Chinese). MR 0194380
Olaf Manz, Arithmetical conditions on character degrees and group structure, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 65–69. MR 933400

W. J. Shi, Characterization of finite simple groups and related topics, Adv. in Math. (China) 20 (1991), 135-141.

Michio Suzuki, Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc. 99 (1961), 425–470. MR 131459, DOI https://doi.org/10.1090/S0002-9947-1961-0131459-5
J. S. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), no. 2, 487–513. MR 617092, DOI https://doi.org/10.1016/0021-8693%2881%2990218-0
Ji Ping Zhang, Finite groups with many conjugate elements, J. Algebra 170 (1994), no. 2, 608–624. MR 1302859, DOI https://doi.org/10.1006/jabr.1994.1356