• $z$-Classes in finite groups of conjugate type ($n$, 1)

• # Fulltext

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

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

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019