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Exponent of a finite group of odd order with an involutory automorphism

  • Sara Rodrigues
  • Pavel ShumyatskyEmail author
Article

Abstract

Let G be a finite group of odd order admitting an involutory automorphism \(\phi \). We obtain two results bounding the exponent of \([G,\phi ]\). Denote by \(G_{-\phi }\) the set \(\{[g,\phi ]\,\vert \, g\in G\}\) and by \(G_{\phi }\) the centralizer of \(\phi \), that is, the subgroup of fixed points of \(\phi \). The obtained results are as follows.
  1. 1.

    Assume that the subgroup \(\langle x,y\rangle \) has derived length at most d and \(x^e=1\) for every \(x,y\in G_{-\phi }\). Suppose that \(G_\phi \) is nilpotent of class c. Then the exponent of \([G,\phi ]\) is (cde)-bounded.

     
  2. 2.

    Assume that \(G_\phi \) has rank r and \(x^e=1\) for each \(x\in G_{-\phi }\). Then the exponent of \([G,\phi ]\) is (er)-bounded.

     

Keywords

Finite groups Automorphisms Rank Exponent 

Mathematics Subject Classification

20D45 

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BrasiliaBrasíliaBrazil

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