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\(\varvec{z}\)-Classes in finite groups of conjugate type \(\varvec{(n,1)}\)


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.

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The authors are thankful to Rahul Kitture for letting them know about his work and for many comments on this work. The authors are grateful to Silvio Dolfi for useful comments and suggestions. This work was part of the MS thesis of Shivam Arora at IISER, Mohali. He gratefully acknowledges the support of IISER Mohali during the course of this work. The second author, Gongopadhyay acknowledges partial support from SERB-DST Grant SR/FTP/MS-004/2010.

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Correspondence to Krishnendu Gongopadhyay.

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Arora, S., Gongopadhyay, K. \(\varvec{z}\)-Classes in finite groups of conjugate type \(\varvec{(n,1)}\). Proc Math Sci 128, 31 (2018).

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  • Conjugacy classes of centralizers
  • z-classes
  • p-groups
  • extraspecial groups

2010 Mathematics Subject Classification

  • Primary: 20D15
  • Secondary: 20E45