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

TL;DR

This paper characterizes certain finite p-groups with maximal z-classes, focusing on groups of conjugate type (n,1), and explores their structure and properties related to centralizers and conjugacy.

Contribution

It provides a characterization of p-groups of conjugate type (n,1) that attain the maximal number of z-classes, extending previous bounds.

Findings

Maximal number of z-classes in non-abelian p-groups is achieved by specific conjugate type groups.

Characterization of p-groups with prime order commutator subgroup and maximal z-classes.

Identification of structural properties of groups reaching the upper bound of z-classes.

Abstract

Two elements in a group $G$ are said to $z$-equivalent or to be in the same $z$-class if their centralizers are conjugate in $G$. In \cite{kkj}, it was proved that a non-abelian $p$-group $G$ can have at most $\frac{p^k-1}{p-1} +1$ number of $z$-classes, where $|G/Z(G)|=p^k$. In this note, 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.