Articles written in Proceedings – Mathematical Sciences

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


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      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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