Advertisement

Archiv der Mathematik

, Volume 106, Issue 4, pp 301–304 | Cite as

On a class of finite capable p-groups

  • Peter SchmidEmail author
Article

Abstract

A group G is called capable if there is a group H such that \({G \cong H/Z(H)}\) is isomorphic to the group of inner automorphisms of H. We consider the situation that G is a finite capable p-group for some prime p. Suppose G has rank \({d(G) \ge 2}\) and Frattini class \({c \ge 1}\), which by definition is the length of a shortest central series of G with all factors being elementary abelian. There is up to isomorphism a unique largest p-group \({G_d^c}\) with rank d and Frattini class c, and G is an epimorphic image of \({G_d^c}\). We prove that this \({G_d^c}\) is capable; more precisely, we have \({G_d^c \cong G_d^{c+1}/Z(G_d^{c+1})}\).

Keywords

Capable groups p-groups Schur multiplier Frattini extension Lie modules 

Mathematics Subject Classification

20D15 20J05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beyl F.R., Felgner U., Schmid P.: On groups occurring as center factor groups. J. Algebra 61, 161–177 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blackburn N., Evens L.: Schur multipliers of p-groups. J. Reine Angew. Math. 309, 100–113 (1979)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bryant R.M., Kovács L.G.: Lie representations and groups of prime power order. J. London Math. Soc. (2) 17, 415–421 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hall P.: The classification of prime power groups. J. Reine Angew. Math. 182, 130–141 (1940)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Huppert B., Blackburn N.: Finite Groups II. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  6. 6.
    Lazard M.: Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. École Norm. Sup. 71, 101–190 (1954)MathSciNetzbMATHGoogle Scholar
  7. 7.
    I.R. Šafarevič (Shafarevich), On the construction of fields with a given Galois goup of order \({\ell^{a}}\), Amer. Math. Soc. Transl. 4 (1956), 107–142.Google Scholar
  8. 8.
    Shafarevich I.R.: Factors of a decreasing central series. Math. Notes 54, 262–264 (1989)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

Personalised recommendations