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      https://www.ias.ac.in/article/fulltext/pmsc/128/03/0031

    • Keywords

       

      Conjugacy classes of centralizers; $z$-classes; $p$-groups; extraspecial groups

    • Abstract

       

      Two elements in a group $G$ are said to be $z$-equivalent or to be in the same $z$-class if their centralizers are conjugate in $G$. In a recent work, Kulkarni et al. (J. Algebra Appl., 15 (2016) 1650131) proved that a non-abelian $p$-group $G$ can have at most $\frac{p^{k}−1}{p−1} + 1$ number of $z$-classes, where $|G/Z(G)| = p^{k}$ . Here, we characterize the $p$-groups of conjugate type ($n$, 1) attaining this maximal number. As a corollary, we characterize $p$-groups having prime order commutator subgroup and maximal number of $z$-classes.

    • Author Affiliations

       

      SHIVAM ARORA1 KRISHNENDU GONGOPADHYAY2

      1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada
      2. Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, S.A.S. Nagar, Punjab 140 306, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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