Infinite Sharply Multiply Transitive Groups

Survey Article

Abstract

The finite sharply 2-transitive groups were classified by Zassenhaus in the 1930’s. They essentially all look like the group of affine linear transformations \(x\mapsto ax+b\) for some field (or at least near-field) \(K\). However, the question remained open whether the same is true for infinite sharply 2-transitive groups. There has been extensive work on the structures associated to such groups indicating that Zassenhaus’ results might extend to the infinite setting. For many specific classes of groups, like Lie groups, linear groups, or groups definable in o-minimal structures it was indeed proved that all examples inside the given class arise in this way as affine groups. However, it recently turned out that the reason for the lack of a general proof was the fact that there are plenty of sharply 2-transitive groups which do not arise from fields or near-fields! In fact, it is not too hard to construct concrete examples (see below). In this note, we survey general sharply \(n\)-transitive groups and describe how to construct examples not arising from fields.

Keywords

Sharply 2-transitive Free product Nearfield 

Mathematics Subject Classification

20B22 

References

  1. 1.
    Borovik, A., Nesin, A.: Groups of Finite Morley Rank. Oxford Logic Guides, vol. 26. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1994) MATHGoogle Scholar
  2. 2.
    Cameron, P.J.: Bases in permutation groups. In: Kaye, R., Macpherson, D. (eds.) Automorphisms of First-Order Structures. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1994) Google Scholar
  3. 3.
    de Medts, T., Weiss, R.: Moufang sets and Jordan division algebras. Math. Ann. 335, 415–433 (2006) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dixon, J.D., Mortimer, B.: Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer, New York (1996) MATHGoogle Scholar
  5. 5.
    Glasner, Y., Gulko, D.: Sharply two transitive linear groups. Int. Math. Res. Not. 2014(10), 2691–2701 (2014) MathSciNetMATHGoogle Scholar
  6. 6.
    Glauberman, G., Mann, A., Segev, Y.: A note on groups generated by involutions and sharply 2-transitive groups. Proc. Am. Math. Soc. 143(5), 1925–1932 (2015) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hall, M.: On a theorem of Jordan. Pac. J. Math. 4, 219–226 (1954) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Jordan, C.: Recherches sur les substitutions. J. Math. Pures Appl. 17, 351–363 (1872) Google Scholar
  9. 9.
    Kerby, W.: On infinite sharply multiply transitive groups. In: Hamburger Mathematische Einzelschriften, Neue Folge, Heft 6. Vandenhoeck & Ruprecht, Göttingen (1974). 71 pp. Google Scholar
  10. 10.
    Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory. Interscience, New York (1966) MATHGoogle Scholar
  11. 11.
    Mayr, P.: Sharply 2-transitive groups with point stabilizer of exponent 3 or 6. Proc. Am. Math. Soc. 134(1), 9–13 (2006) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Mazurov, V.D., Khukhro, E.I. (eds.): The Kourovka Notebook. Unsolved Problems in Group Theory 18 (2014). arXiv:1401.0300 MATHGoogle Scholar
  13. 13.
    Neumann, B.H.: On the commutativity of addition. J. Lond. Math. Soc. 15, 203–208 (1940) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rips, E., Tent, K.: Sharply 2-transitive group in characteristic 0. Preprint Google Scholar
  15. 15.
    Rips, E., Segev, Y., Tent, K.: A sharply 2-transitive group without a non-trivial abelian normal subgroup. J. Eur. Math. Soc. (2016, to appear) Google Scholar
  16. 16.
    Tent, K.: Sharply \(n\)-transitive groups in o-minimal structures. Forum Math. 12(1), 65–75 (2000) MathSciNetMATHGoogle Scholar
  17. 17.
    Tent, K.: Sharply 3-transitive groups. Adv. Math. 286, 722–728 (2016) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Tent, K., Ziegler, M.: Sharply 2-transitive groups. Adv. Geom. 16(1), 131–134 (2016) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Tits, J.: Groupes triplement transitifs et generalisations. Algebre et Theorie de nombres. Colloq. Int. Cent. Natl. Rech. Sci. 24, 207–208 (1950) MathSciNetMATHGoogle Scholar
  20. 20.
    Tits, J.: Sur les groupes doublement transitifs continus. Comment. Math. Helv. 26, 203–224 (1952) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Türkelli, S.: Splitting of sharply 2-transitive groups of characteristic 3. Turk. J. Math. 28(3), 295–298 (2004) MathSciNetMATHGoogle Scholar
  22. 22.
    Wähling, H.: Lokal endliche, scharf zweifach transitive Permutationsgruppen. Abh. Math. Semin. Univ. Hamb. 56, 107–113 (1986) (German) [Locally finite, sharply doubly transitive permutation groups] MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zassenhaus, H.: Über endliche Fastkörper. Abh. Math. Semin. Univ. Hamb. 11, 187–220 (1936) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zassenhaus, H.: Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen. Abh. Math. Semin. Univ. Hamb. 11, 17–40 (1936) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität MünsterMünsterGermany

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