Normal Subgroups of Groups of Prime-Power Order

  • Bruce W. King
Part of the Lecture Notes in Mathematics book series (LNM, volume 372)


In [5] Gaschütz noted that the dihedral and quaternion groups of order 8 cannot be Frattini subgroups of 2-groups. This, together with results in [6], raised the question of what limitations there may be on the structure of those normal subgroups which are contained in the Frattini subgroup of a p-group. Partial answers have been obtained in [l, 2, 3, 6, 7]. It has been shown that the subgroups considered cannot have cyclic centre, and that many such subgroups, if they have two generators, must be metacyclic. It is of interest to remove the restriction that the embedded normal subgroups have two generators. Under some weaker restrictions it may be shown that often such a subgroup N has the property N′ ≤ N P , or N′ ≤ N 4 if p = 2. The parallel results obtained when the embedded normal subgroup has two generators strengthen the results mentioned above.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Homer Bechtell, “Frattini subgroups and O-central groups”, Panifie J. Math. 18 (1966), 15–23. MR33#5725.MathSciNetMATHGoogle Scholar
  2. [2]
    Я. Г. Беркаовии [Ja.G. Berkovič], “Нормальные подгруппы в конечной группе” [Normal subgroups in a finite group], Dokl. Akad. Sauk SSSR 182 (1968), 247–250; Soviet Math. Dokl. 9 (1968), 1117–1120. MR38#4562.Google Scholar
  3. [3]
    N. Blackburn, “On prime-power groups in which the derived group has two generators”, Proc. Cambridge Philos. Soc. 53 (1957), 19–27. MR18,464.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    N. Blackburn, “On prime-power groups with two generators”, Proc. Cambridge Philos. Soc. 54(1958), 327–337. MR21f1348.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Wolfgang Gaschütz, “Über die 0-Untergruppe endlicher Gruppen”, Math. Z. 58 (1953), 160–170. MR15,285.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Charles Hobby, “The Frattini subgroup of a p-group“, Pacific J. Math. 10 (1960), 209–212. MJ122#4780.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Charles R. Hobby, “Generalizations of a theorem of N. Blackburn on p-groups”, Illinois J. Math. 5 (1961), 225–227. MR23#A209.MathSciNetMATHGoogle Scholar
  8. [8]
    Bertram Huppert, “Über das produkt von paarweise vertauschbaren zyklischen Gruppen”, Math. Z. 58 (1953), 243–264. MR14,1059 and MR17,1436.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    B. Ruppert, Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, Band 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967). MR37#302.Google Scholar
  10. [10]
    Bruce W. King, “Presentations of metacyclic groups”, Bull. Austral. Math. Soc. 8 (1973), 103–131. 261.245.20016.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Bruce W. King
    • 1
  1. 1.Riverina College of Advanced EducationWagga WaggaAustralia

Personalised recommendations