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On \({{\varvec{n}}}\)-th class preserving automorphisms of \({{\varvec{n}}}\)-isoclinism family

  • Surjeet KourEmail author
Article
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Abstract

Let G be a finite group and let M and N be two normal subgroups of G. Let \(\hbox {Aut}_N^M(G)\) denote the group of all automorphisms of G which fix N element-wise and act trivially on G / M. Let n be a positive integer. In this article, we have shown that if G and H are two n-isoclinic groups, then there exists an isomorphism from \(\hbox {Aut}_{Z_n(G)}^{\gamma _{n+1}(G)}(G)\) to \(\hbox {Aut}_{Z_n(H)}^{\gamma _{n+1}(H)}(H)\), which maps the group of n-th class preserving automorphisms of G to the group of n-th class preserving automorphisms of H. Also, for a nilpotent group G of class \((n+1)\), if \(\gamma _{n+1}(G)\) is cyclic, then we prove that \(\hbox {Aut}_{Z_n(G)}^{\gamma _{n+1}(G)}(G)\) is isomorphic to the group of inner automorphisms of a quotient group of G.

Keywords

Finite group inner automorphism n-isoclinism n-th class preserving automorphism 

2010 Mathematics Subject Classification

20D15 20D45 

Notes

Acknowledgements

The author would like to thank the referee/referees for their valuable suggestions. This research is partially supported by SERB-DST Grant YSS/2015/001567.

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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of TechnologyNew DelhiIndia

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