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Thompson-like characterization of solubility for products of finite groups

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Abstract

A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson’s theorem from the perspective of factorized groups. More precisely, we study finite groups \(G = AB\) with subgroups \(A,\, B\) such that \(\langle a, b\rangle \) is soluble for all \(a \in A\) and \(b \in B\). In this case, the group G is said to be an \({{\mathcal {S}}}\)-connected product of the subgroups A and B for the class \({\mathcal {S}}\) of all finite soluble groups. Our Main Theorem states that \(G = AB\) is \({\mathcal {S}}\)-connected if and only if [A, B] is soluble. In the course of the proof, we derive a result about independent primes regarding the soluble graph of almost simple groups that might be interesting in its own right.

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References

  1. Abe, S., Iiyori, N.: A generalization of prime graphs of finite groups. Hokkaido Math. J. 29, 391–407 (2000)

    Article  MathSciNet  Google Scholar 

  2. Amberg, B., Carocca, A., Kazarin, L.: Criteria for the solubility and non-simplicity of finite groups. J. Algebra 285, 58–72 (2005)

    Article  MathSciNet  Google Scholar 

  3. Amberg, B., Franciosi, S., de Giovanni, F.: Products of Groups. Clarendon Press, Oxford (1992)

    MATH  Google Scholar 

  4. Amberg, B., Kazarin, L.: On the soluble graph of a finite simple group. Commun. Algebra 41, 2297–2309 (2013)

    Article  MathSciNet  Google Scholar 

  5. Asaad, M., Shaalan, A.: On the supersolvability of finite groups. Arch. Math. 53, 318–326 (1989)

    Article  MathSciNet  Google Scholar 

  6. Baer, R.: Supersoluble immersion. Can. J. Math. 11, 353–369 (1959)

    Article  MathSciNet  Google Scholar 

  7. Ballester-Bolinches, A., Ezquerro, L.M.: Classes of Finite Groups. Springer, Berlin (2006)

    MATH  Google Scholar 

  8. Ballester-Bolinches, A., Pedraza-Aguilera, M.C.: On finite soluble products of ${\cal{N}}$-connected groups. J. Group Theory 2, 291–299 (1999)

    Article  MathSciNet  Google Scholar 

  9. Beidleman, J., Heineken, H.: Pairwise ${\cal{N}}$-connected products of certain classes of finite groups. Commun. Algebra 32, 4741–4752 (2004)

    Article  MathSciNet  Google Scholar 

  10. Bray, J.N., Holt, D.F., Roney-Dougal, C.M.: The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  11. Carocca, A.: A note on the product of ${mathcal F }$-subgroups in a finite group. Proc. Edinb. Math. Soc. (2) 39, 37–42 (1996)

    Article  MathSciNet  Google Scholar 

  12. Carocca, A.: Solvability of factorized finite groups. Glasgow Math. J. 42, 271–274 (2000)

    Article  MathSciNet  Google Scholar 

  13. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)

    MATH  Google Scholar 

  14. Dixon, J.D., Mortimer, B.: Permutation Groups. Springer, Berlin (1996)

    Book  Google Scholar 

  15. Doerk, K., Hawkes, T.: Finite Soluble Groups. Walter de Gruyter, Berlin (1992)

    Book  Google Scholar 

  16. Dolfi, S., Guralnick, R.M., Herzog, M., Praeger, C.E.: A new solvability criterion for finite groups. J. Lond. Math. Soc. 85, 269–281 (2012)

    Article  MathSciNet  Google Scholar 

  17. Gállego, M. P., Hauck, P., Kazarin, L. S., Martínez-Pastor, A., Pérez-Ramos, M. D.: Products of finite connected subgroups. Preprint. arXiv:1908.03347

  18. Gállego, M.P., Hauck, P., Pérez-Ramos, M.D.: Soluble products of connected subgroups. Rev. Mat. Iberoam. 24, 433–461 (2008)

    Article  MathSciNet  Google Scholar 

  19. Gállego, M.P., Hauck, P., Pérez-Ramos, M.D.: On 2-generated subgroups and products of groups. J. Group Theory 11, 851–867 (2008)

    Article  MathSciNet  Google Scholar 

  20. Gállego, M.P., Hauck, P., Pérez-Ramos, M.D.: Saturated formations and products of connected subgroups. J. Algebra 333, 105–119 (2011)

    Article  MathSciNet  Google Scholar 

  21. Gállego, M.P., Hauck, P., Pérez-Ramos, M.D.: 2-Engel relations between subgroups. J. Algebra 447, 31–55 (2016)

    Article  MathSciNet  Google Scholar 

  22. Gordeev, N., Grunewald, F., Kunyavskiĭ, B., Plotkin, E.: Baer–Suzuki theorem for the solvable radical of a finite group. C. R. Acad. Sci. Paris Sér. I(347), 217–222 (2009)

    Article  MathSciNet  Google Scholar 

  23. Gordeev, N., Grunewald, F., Kunyavskiĭ, B., Plotkin, E.: From Thompson to Baer–Suzuki: a sharp characterization of the solvable radical. J. Algebra 323, 2888–2904 (2010)

    Article  MathSciNet  Google Scholar 

  24. Guest, S.: A solvable version of the Baer–Suzuki theorem. Trans. Am. Math. Soc. 362, 5909–5946 (2010)

    Article  MathSciNet  Google Scholar 

  25. Guest, S., Levy, D.: Criteria for solvable radical membership via $p$-elements. J. Algebra 415, 88–111 (2014)

    Article  MathSciNet  Google Scholar 

  26. Guralnick, R., Kunyavskiĭ, B., Plotkin, E., Shalev, A.: Thompson-like characterizations of the solvable radical. J. Algebra 300, 363–375 (2006)

    Article  MathSciNet  Google Scholar 

  27. Grunewald, F., Kunyavskiĭ, B., Plotkin, E.: Characterization of solvable groups and solvable radical. Int. J. Algebra Comput. 23, 1011–1062 (2013)

    Article  MathSciNet  Google Scholar 

  28. Hauck, P., Martínez-Pastor, A., Pérez-Ramos, M.D.: Products of $\cal{N}$-connected groups. Ill. J. Math. 47, 1033–1045 (2003)

    Article  MathSciNet  Google Scholar 

  29. Hering, C., Liebeck, M.W., Saxl, J.: The factorizations of the finite exceptional groups of Lie type. J. Algebra 106, 517–527 (1987)

    Article  MathSciNet  Google Scholar 

  30. Huppert, B.: Zweifach transitive, auflösbare Permutationsgruppen. Math. Z. 68, 126–150 (1957)

    Article  MathSciNet  Google Scholar 

  31. Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)

    Book  Google Scholar 

  32. Iiyory, N.: $p$-Solvability and a generalization of prime graphs of finite groups. Commun. Algebra 30, 1679–1691 (2002)

    Article  MathSciNet  Google Scholar 

  33. Kleidman, P., Liebeck, M.: The subgroup structure of the finite classical groups. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  34. Liebeck, M.W., Praeger, C.E., Saxl, J.: The maximal factorizations of the finite simple groups and their automorphism groups. Mem. AMS 86, 432 (1990)

    MathSciNet  MATH  Google Scholar 

  35. Maier, R.: A completeness property of certain formations. Bull. Lond. Math. Soc. 24, 540–544 (1992)

    Article  MathSciNet  Google Scholar 

  36. Malle, G., Saxl, J., Weigel, T.: Generation of classical groups. Geom. Dedicata 49, 85–116 (1994)

    Article  MathSciNet  Google Scholar 

  37. Ramanujan, S.: A proof of Bertrand’s postulate. J. Indian Math. Soc. 11, 181–182 (1919)

    Google Scholar 

  38. The GAP Group, GAP—Groups, Algorithms, and Programming. http://www.gap-system.org, Version 4.10.2 (2019)

  39. Thompson, J.: Nonsolvable finite groups all of whose local subgroups are solvable. Bull. Am. Math. Soc. 74, 383–437 (1968)

    Article  MathSciNet  Google Scholar 

  40. Wiegold, J., Williamson, A.G.: The factorisation of the alternating and symmetric groups. Math. Z. 175, 171–179 (1980)

    Article  MathSciNet  Google Scholar 

  41. Zorn, M.: Nilpotency of finite groups. Bull. Am. Math. Soc. 42, 485–486 (1936)

    MATH  Google Scholar 

  42. Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3, 265–284 (1892)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Research supported by Proyectos PROMETEO/2017/057 from the Generalitat Valenciana (Valencian Community, Spain) and PGC2018-096872-B-I00 from the Ministerio de Ciencia, Innovación y Universidades, Spain, and FEDER, European Union; and second author also by Project VIP-008 of Yaroslavl P. Demidov State University.

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Authors and Affiliations

  1. Fachbereich Informatik, Universität Tübingen, Sand 13, 72076, Tübingen, Germany

    P. Hauck

  2. Department of Mathematics, Yaroslavl P. Demidov State University, Sovetskaya Str 14, 150014, Yaroslavl, Russia

    L. S. Kazarin

  3. Instituto Universitario de Matemática Pura y Aplicada (IUMPA-UPV), Universitat Politècnica de València, Camino de Vera s/n, 46022, Valencia, Spain

    A. Martínez-Pastor

  4. Departament de Matemàtiques, Universitat de València, C/ Doctor Moliner 50, 46100, Burjassot (València), Spain

    M. D. Pérez-Ramos

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  1. P. Hauck
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  2. L. S. Kazarin
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  3. A. Martínez-Pastor
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  4. M. D. Pérez-Ramos
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Correspondence to A. Martínez-Pastor.

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Dedicated to the memory of M. Pilar Gállego (1958–2019).

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Hauck, P., Kazarin, L.S., Martínez-Pastor, A. et al. Thompson-like characterization of solubility for products of finite groups. Annali di Matematica 200, 337–362 (2021). https://doi.org/10.1007/s10231-020-00998-z

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  • DOI: https://doi.org/10.1007/s10231-020-00998-z

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