JournalsrsmupVol. 142falsepp. 69–80

Groups with few self-centralizing subgroups which are not self-normalizing

  • Mahmoud Hassanzadeh

    Iran University of Science & Technology, Tehran, Iran
  • Zohreh Mostaghim

    Iran University of Science & Technology, Tehran, Iran
Groups with few self-centralizing subgroups which are not self-normalizing cover

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Abstract

A self-normalizing subgroup is always self-centralizing, but the converse is not necessarily true. Given a finite group GG, we denote by w(G)w(G) the number of all self-centralizing subgroups of GG which are not self-normalizing. We observe that w(G)=0w(G) = 0 if and only if GG is abelian, and that if GG is nonabelian nilpotent then w(G)3w(G)\geq 3. We also prove that if w(G)20w(G)\leq 20 then GG is solvable. Finally, we provide structural information in the case when w(G)3w(G)\leq 3.

Cite this article

Mahmoud Hassanzadeh, Zohreh Mostaghim, Groups with few self-centralizing subgroups which are not self-normalizing. Rend. Sem. Mat. Univ. Padova 142 (2019), pp. 69–80

DOI 10.4171/RSMUP/30