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\({\varvec{z}}\)-classes in groups: a survey

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Abstract

This survey article explores the notion of z-classes in groups. The concept introduced here is related to the notion of orbit types in transformation groups, and types or genus in the representation theory of finite groups of Lie type. Two elements in a group are said to be z-equivalent (or z-conjugate) if their centralizers are conjugate. This is a weaker notion than the conjugacy of elements. In this survey article, we present several known results on this topic and suggest some further questions.

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Acknowledgements

The second named author gratefully acknowledges the opportunity to attend a series of lectures in his graduate days given at HRI by Ravi Kulkarni, introducing the notion of z-classes. The authors would like to thank the referees for their helpful comments.

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Authors and Affiliations

  1. IISER Mohali, Knowledge City, Sector 81, S.A.S. Nagar, Punjab, 140306, India

    Sushil Bhunia

  2. IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune, 411 008, India

    Anupam Singh

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  1. Sushil Bhunia
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  2. Anupam Singh
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Additional information

Communicated by Gadadhar Misra.

Dedicated to Professor I. B. S. Passi on the occasion of his $$80^{th}$$ 80 th birthday.

The second named author would like to acknowledge support of SERB core research grant CRG/2019/000271 for this work. This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participating in the program - Group Algebras, Representations and Computation ICTS/Prog-garc2019/10.

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Bhunia, S., Singh, A. \({\varvec{z}}\)-classes in groups: a survey. Indian J Pure Appl Math 52, 713–720 (2021). https://doi.org/10.1007/s13226-021-00186-6

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  • DOI: https://doi.org/10.1007/s13226-021-00186-6

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