Archiv der Mathematik

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

On a class of finite capable p-groups

  • Peter SchmidEmail author


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 \({G_d^c \cong G_d^{c+1}/Z(G_d^{c+1})}\).


Capable groups p-groups Schur multiplier Frattini extension Lie modules 

Mathematics Subject Classification

20D15 20J05 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

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