We prove that the nilpotent product of a set of groups $A_1, \ldots , A_s$ has finite palindromic width if and only if the palindromic widths of $A_i$, $i = 1, \ldots , s$, are finite. We give a new proof that the commutator width of $F_n \wr K$ is infinite, where $F_n$ is a free group of rank $n\geq 2$ and $K$ is a finite group. This result, combining with a result of Fink [9] gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set.

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.

In this paper, we answer questions raised by Bardakov and Gongopadhyay (Commun. Algebra 43(11) (2015) 4809–4824). We prove that the palindromic width of HNN extension of a group by proper associated subgroups is infinite. We also prove that the palindromic width of the amalgamated free product of two groups via a proper subgroup is infinite (except when the amalgamated subgroup has index two in each of the factors). Combining these results it follows that the palindromic width of the fundamental group of a graph of groups is mostly infinite.