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On the solvability of regular subgroups in the holomorph of a finite solvable group

    We exhibit infinitely many natural numbers n for which there exists at least one insolvable group of order n, and yet the holomorph of every solvable group of order n has no insolvable regular subgroup. We also solve Problem 19.90(d) in the Kourovka notebook.

    Communicated by E. O’Brien

    AMSC: 20B35, 20F16, 20D05


    • 1. H. U. Besche, B. Eick and E. A. O’Brien, A millennium project: Constructing small groups, Int. J. Algebra Comput. 12(5) (2002) 623–644. Link, ISIGoogle Scholar
    • 2. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997) 23–265. Crossref, ISIGoogle Scholar
    • 3. R. Brauer, On simple groups of order 5 3a 2b, Bull. Amer. Math. Soc. 74 (1968) 900–903. Crossref, ISIGoogle Scholar
    • 4. N. P. Byott, Hopf-Galois structures on field extensions with simple Galois groups, Bull. London Math. Soc. 36(1) (2004) 23–29. CrossrefGoogle Scholar
    • 5. N. P. Byott, Solubility criteria for Hopf-Galois structures, New York J. Math. 21 (2015) 883–903. ISIGoogle Scholar
    • 6. L. N. Childs, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, Mathematical Surveys and Monographs, Vol. 80 (American Mathematical Society, Providence, RI, 2000). CrossrefGoogle Scholar
    • 7. R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comput. 66(217) (1997) 433–449. Crossref, ISIGoogle Scholar
    • 8. H. Dietrich and B. Eick, On the groups of cube-free order, J. Algebra 292(1) (2005) 122–137; Addendum, 367 (2012) 247–248. Crossref, ISIGoogle Scholar
    • 9. J. Douglas, On the supersolvability of bicyclic groups, Proc. Natl. Acad. Sci. USA 47 (1961) 1493–1495. Crossref, ISIGoogle Scholar
    • 10. The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.10.2 (2019), Google Scholar
    • 11. L. Guarnieri and L. Vendramin, Skew braces and the Yang–Baxter equation, Math. Comput. 86(307) (2017) 2519–2534. Crossref, ISIGoogle Scholar
    • 12. M. Hall, Jr., The Theory of Groups (The Macmillan Co., New York, 1959). Google Scholar
    • 13. M. Herzog, On finite simple groups of order divisible by three primes only, J. Algebra 10 (1968) 383–388. Crossref, ISIGoogle Scholar
    • 14. I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, Vol. 92 (American Mathematical Society, Providence, RI, 2008). CrossrefGoogle Scholar
    • 15. N. Ito, Über das Produkt von zwei abelschen Gruppen, Math. Z. 62 (1955) 400–401. CrossrefGoogle Scholar
    • 16. O. H. Kegel, Produkte nilpotenter Gruppen, Arch. Math. (Basel) 12 (1961) 90–93. CrossrefGoogle Scholar
    • 17. E. Khukhro and V. Mazurov, Unsolved Problems in Group Theory, Kourovka Notebook, No. 19 (2019), Google Scholar
    • 18. OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, Google Scholar
    • 19. A. Smoktunowicz and L. Vendramin, On skew braces (with an appendix by N. Byott and L. Vendramin), J. Comb. Algebra 2(1) (2018) 47–86. Crossref, ISIGoogle Scholar
    • 20. M. Suzuki, A new type of simple groups of finite order, Proc. Nat. Acad. Sci. U.S.A. 46 (1960) 868–870. Crossref, ISIGoogle Scholar
    • 21. M. Suzuki, Group Theory. I, Grundlehren der Mathematischen Wissenschaften, Vol. 247 (Springer-Verlag, Berlin-New York, 1982). CrossrefGoogle Scholar
    • 22. C. Tsang, Galois module structures and Hopf-Galois structures on extensions of number fields, Postdoctoral Report, Tsinghua University (2018). Google Scholar
    • 23. C. Tsang, Non-existence of Hopf-Galois structures and bijective crossed homomorphisms, J. Pure Appl. Algebra 223(7) (2019) 2804–2821. Crossref, ISIGoogle Scholar
    • 24. C. Tsang, Hopf-Galois structures on a Galois Sn-extension, J. Algebra 531(1) (2019) 349–361. CrossrefGoogle Scholar
    • 25. D. Wales, Simple groups of order 7 3a 2b, J. Algebra 16 (1970) 575–596. Crossref, ISIGoogle Scholar
    • 26. R. A. Wilson, The Finite Simple Groups, Graduate Texts in Mathematics, Vol. 251 (Springer-Verlag, London, 2009). CrossrefGoogle Scholar
    Published: 22 October 2019
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