On the number of isomorphism classes of derived subgroups

  • Leyli Jafari Taghvasani
  • Soran Marzang
  • Mohammad ZarrinEmail author


We show that a finite nonabelian characteristically simple group G satisfies n = |π(G)| + 2 if and only if GA5, where n is the number of isomorphism classes of derived subgroups of G and π(G) is the set of prime divisors of the group G. Also, we give a negative answer to a question raised in M. Zarrin (2014).


derived subgroup simple group 




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  1. [1]
    M. J. J. Barry, M. B. Ward: Simple groups contain minimal simple groups. Publ. Mat., Barc. 41 (1997), 411–415.MathSciNetCrossRefGoogle Scholar
  2. [2]
    The GAP Group: GAP-Groups, Algorithms, Programming-a System for Computa-tional Discrete Algebra, Version 4. 4. 2005; Available at http://www. gap-system. org. swGoogle Scholar
  3. [3]
    F. de Giovanni, D. J. S. Robinson: Groups with finitely many derived subgroups. J. Lond. Math. Soc., II. Ser. 71 (2005), 658–668.MathSciNetCrossRefGoogle Scholar
  4. [4]
    M. Herzog: On finite simple groups of order divisible by three primes only. J. Algebra 10 (1968), 383–388.MathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Herzog, P. Longobardi, M. Maj: On the number of commutators in groups. Ischia Group Theory 2004 (Z. Arad et al., eds.). Contemporary Mathematics 402, American Mathematical Society, Providence, 2006, pp. 181–192.zbMATHGoogle Scholar
  6. [6]
    B. Huppert: Finite Groups I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134, Springer, Berlin, 1967. (In German.)Google Scholar
  7. [7]
    M. W. Liebeck, E. A. O’Brien, A. Shalev, P. H. Tiep: The Ore conjecture. J. Eur. Math. Soc. (JEMS) 12 (2010), 939–1008.MathSciNetCrossRefGoogle Scholar
  8. [8]
    P. Longobardi, M. Maj, D. J. S. Robinson: Locally finite groups with finitely many iso-morphism classes of derived subgroups. J. Algebra 393 (2013), 102–119.MathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Longobardi, M. Maj, D. J. S. Robinson, H. Smith: On groups with two isomorphism classes of derived subgroups. Glasg. Math. J. 55 (2013), 655–668.MathSciNetCrossRefGoogle Scholar
  10. [10]
    W. Shi: On simple K4-groups. Chin. Sci. Bull. 36 (1991), 1281–1283.Google Scholar
  11. [11]
    M. Zarrin: On groups with finitely many derived subgroups. J. Algebra Appl. 13 (2014), Article ID 1450045, 5 pages.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  • Leyli Jafari Taghvasani
    • 1
  • Soran Marzang
    • 1
  • Mohammad Zarrin
    • 1
    Email author
  1. 1.Department of MathematicsUniversity of KurdistanSanandajIran

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