Semi-extraspecial Groups

  • Mark L. LewisEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 277)


We survey the results regarding semi-extraspecial p-groups. Semi-extraspecial groups can be viewed as generalizations of extraspecial groups. We present the connections between semi-extraspecial groups, Camina groups, and VZ-groups, and give upper bounds on the order of the center and the orders of abelian normal subgroups. We define ultraspecial groups to be semi-extraspecial groups where the center is as large as possible, and demonstrate a connection between ultraspecial groups that have at least two abelian subgroups whose order is the maximum and semifields.


p-group Extraspecial group Semifields 

Mathematics Subject Classification 2010:



  1. 1.
    Albert, A.A.: On nonassociative division algebras. Trans. Am. Math. Soc. 72, 296–309 (1952)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Albert, A.A.: Finite noncommutative division algebras. Proc. Am. Math. Soc. 9, 928–932 (1958)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Albert, A.A.: Isotopy for generalized twisted fields. An. Acad. Brasil. Ci. 33, 265–275 (1961)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beisiegel, B.: Semi-extraspezielle \(p\)-Gruppen. Math. Z. 156, 247–254 (1977)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symbolic Comput. 24, 235–265 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chillag, D., MacDonald, I.D.: Generalized Frobenius groups. Israel J. Math. 47, 111–122 (1984)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Combarro, E.F., Rúa, I.F., Ranilla, J.: New advances in the computational exploration of semifields. Int. J. Comput. Math. 88, 1990–2000 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Combarro, E.F., Rúa, I.F., Ranilla, J.: Finite semifields with \(7^4\) elements. Int. J. Comput. Math. 89, 1865–1878 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cordero, M., Wene, G.P.: A survey of finite semifields. Discrete Math. 208(209), 125–137 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cronheim, A.: \(T\)-groups and their geometry. Illinois J. Math. 9, 1–30 (1965)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dark, R., Scoppola, C.M.: On Camina groups of prime power order. J. Algebra 181, 787–802 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dempwolff, U.: Semifield planes of order \(81\). J. Geom. 89, 1–16 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dickson, L.E.: Linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc. 7, 370–390 (1906)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dickson, L.E.: On commutative linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc. 7, 514–522 (1906)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dolfi, S., Moretó, A., Navarro, G.: The groups with exactly one class of size a multiple of \(p\). J. Group Theory 12, 219–234 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fernández-Alcober, G.A., Moretó, A.: Groups with two extreme character degrees and their normal subgroups. Trans. Am. Math. Soc. 353, 2171–2192 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Goldstein, D., Guralnick, R.M., Lewis, M.L., Moretó, A., Navarro, G., Tiep, P.H.: Groups with exactly one irreducible character of degree divisible by \(p\). Algebra Number Theory 8, 397–428 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hall, P.: The classification of prime-power groups. J. Reine Angew. Math. 182, 130–141 (1940)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Heineken, H.: Nilpotente Gruppen, deren sämtliche Normalteiler charakteristisch sind, Arch. Math. (Basel) 33(1979/80), 497–503Google Scholar
  20. 20.
    Hiramine, Y.: Automorphisms of \(p\)-groups of semifield type. Osaka J. Math. 20, 735–746 (1983)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Huppert, B.: Endliche Gruppen I. Springer, Berlin, New York (1967)CrossRefGoogle Scholar
  22. 22.
    Huppert, B.: Character Theory of Finite Groups. Walter de Gruyter & Co., Berlin (1998)CrossRefGoogle Scholar
  23. 23.
    Isaacs, I.M.: Finite Group Theory. American Mathematical Society, Providence, RI (2008)zbMATHGoogle Scholar
  24. 24.
    Isaacs, I.M., Lewis, M.L.: Camina \(p\)-groups that are generalized Frobenius complements. Arch. Math. (Basel) 104, 401–405 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kantor, W.M.: Finite semifields in finite geometries, groups, and computation. Walter de Gruyter GmbH & Co. KG, Berlin, pp. 103–114 (2006)Google Scholar
  26. 26.
    Kleinfeld, E.: Techniques for enumerating Veblen-Wedderburn systems. J. Assoc. Comput. Mach. 7, 330–337 (1960)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Knarr, N., Stroppel, M.J.: Heisenberg groups, semifields, and translation planes. Beitr. Algebra Geom. 56, 115–127 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Knuth, D.E.: Finite semifields and projective planes. J. Algebra 2, 182–217 (1965)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Knuth, D.E.: A class of projective planes. Trans. Am. Math. Soc. 115, 541–549 (1965)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Lewis, M.L.: Character tables of groups where all nonlinear irreducible characters vanish off the center in Ischia group theory 2008, 174–182. World Scientific Publishing, Hackensack, NJ (2009)Google Scholar
  31. 31.
    Lewis, M.L.: Brauer pairs of Camina \(p\)-groups of nilpotence class \(2\). Arch. Math. (Basel) 92, 95–98 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lewis, M.L.: Classifying Camina groups: a theorem of Dark and Scoppola. Rocky Mountain J. Math. 44, 591–597 (2014). Erratum on “Classifying Camina groups: a theorem of Dark and Scoppola” [MR3240515], Rocky Mountain J. Math. 45, 273 (2015)Google Scholar
  33. 33.
    Lewis, M.L.: Centralizers of Camina groups with nilpotence class \(3\). J. Group TheoryGoogle Scholar
  34. 34.
    Lewis, M.L., Moretó, A., Wolf, T.R.: Non-divisibility among character degrees. J. Group Theory 8, 561–588 (2005)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lewis, M.L., Wilson, J.B.: Isomorphism in expanding families of indistinguishable groups. Groups Complex. Cryptol. 4, 73–110 (2012)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Lewis, M.L., Wynn, C.: Supercharacter theories of semiextraspecial \(p\)-groups and Frobenius groups (Submitted)Google Scholar
  37. 37.
    Long, F.W.: Corrections to Dicksons table of three dimensional division algebras over \(F_5\). Math. Comp. 31, 1031–1033 (1977)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Macdonald, I.D.: Some \(p\)-groups of Frobenius and extraspecial type. Israel J. Math. 40, 350–364 (1981)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Macdonald, I.D.: More on \(p\)-groups of Frobenius type. Israel J. Math. 56, 335–344 (1986)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Mann, A.: Some finite groups with large conjugacy classes. Israel J. Math. 71, 55–63 (1990)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Mann, A., Scoppola, C.M.: On \(p\)-groups of Frobenius type. Arch. der Math. 56, 320–332 (1991)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Menichetti, G.: On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field. J. Algebra 47, 400–410 (1977)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Nenciu, A.: Brauer pairs of VZ-groups. J. Algebra Appl. 7, 663–670 (2008)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Newman, M.F., O’Brien, E.A., Vaughan-Lee, M.R.: Groups and nilpotent Lie rings whose order is the sixth power of a prime. J. Algebra 278, 383–401 (2004)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Rocco, N.R., Rocha, J.S.: A note on finite semifields and certain \(p\)-groups of class \(2\). Discrete Math. 275, 355–362 (2004)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Rúa, I.F., Combarro, E.F., Ranilla, J.: Classification of semifields of order \(64\). J. Algebra 322, 4011–4029 (2009)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Rúa, I.F., Combarro, E.F.: Commutative semifields of order \(3^5\). Commun. Algebra 40, 988–996 (2012)CrossRefGoogle Scholar
  48. 48.
    Rúa, I.F., Combarro, E.F., Ranilla, J.: Determination of division algebras with \(243\) elements. Finite Fields Appl. 18, 1148–1155 (2012)MathSciNetCrossRefGoogle Scholar
  49. 49.
    van der Waall, R.W., Kuisch, E.B.: Homogeneous character induction II. J. Algebra 170, 584–595 (1994)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Verardi, L.: Gruppi semiextraseciali di esponente \(p\). Ann. Mat. Pura Appl. 148, 131–171 (1987)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Walker, R.J.: Determination of division algebras with \(32\) elements. Proc. Sympos. Appl. Math. 15, 83–85 (1963)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Warfield Jr. R.B.: Nilpotent groups. In: Lecture Notes in Mathematics, vol. 513. Springer, Berlin, New York (1976)Google Scholar
  53. 53.
    Wynn, C.W.: Supercharacter theories of Camina pairs. Ph.D. Dissertation, Kent State University (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA

Personalised recommendations