Glasnik Matematicki, Vol. 49, No. 1 (2014), 105-111.

FINITE GROUPS HAVING AT MOST 27 NON-NORMAL PROPER SUBGROUPS OF NON-PRIME-POWER ORDER

Jiangtao Shi and Cui Zhang

School of Mathematics and Information Science, Yantai University, Yantai 264005, China
e-mail: jiangtaoshi@126.com

Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Lyngby, Denmark
e-mail: cuizhang2008@gmail.com

Abstract.   We prove that any finite group having at most 27 non-normal proper subgroups of non-prime-power order is solvable except for G≅ A5, the alternating group of degree 5.

2010 Mathematics Subject Classification.   20D10.

Key words and phrases.   Finite group, solvable group, subgroup of non-prime-power order.

Full text (PDF) (free access)

DOI: 10.3336/gm.49.1.08

References:

  1. J. H. Conway, R. T. Curtis, S. P. Norton, et al., Atlas of finite groups, Clarendon Press, Oxford, 1985.
    MathSciNet    

  2. L. E. Dickson, Linear groups with an exposition of the Galois field theory, Leipzig, Teubner, 1901.

  3. J. A. Gallian and D. Moulton, The classification of minimal non-prime-power order groups, Arch. Math. (Basel) 59 (1992), 521-524.
    MathSciNet     CrossRef

  4. G. Pazderski, Über maximale Untergruppen endlicher Gruppen, Math. Nachr. 26 (1963/1964), 307-319.
    MathSciNet    

  5. D. J. S. Robinson, A course in the theory of groups, Springer-Verlag, New York, 1996.
    MathSciNet     CrossRef

  6. J. Shi and C. Zhang, Finite groups with given quantitative non-nilpotent subgroups, Comm. Algebra 39 (2011), 3346-3355.
    MathSciNet     CrossRef

  7. J. Shi and C. Zhang, Finite groups in which the number of classes of non-nilpotent proper subgroups of the same order is given, Chinese Ann. Math. Ser. A 32 (2011), 687-692 (in Chinese).
    MathSciNet    

  8. J. Shi, C. Zhang and D. Liang, Finite groups with some subgroups of given order, Algebra Colloq. 20 (2013), 457-462.
    MathSciNet     CrossRef

  9. M. Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105-145.
    MathSciNet     CrossRef

  10. J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383-437.
    MathSciNet     CrossRef

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