  We discuss the finiteness of the topological entropy of continuous
endomorphims for some classes of locally compact groups. Firstly, we focus on
the abelian case, imposing the condition of being compactly generated, and note
an interesting behaviour of slender groups. Secondly, we remove the condition
of being abelian and consider nilpotent periodic locally compact $p$-groups
($p$ prime), reducing the computations to the case of Sylow $p$-subgroups.
Finally, we investigate locally compact Heisenberg $p$-groups
$\mathbb{H}_{n}(\mathbb{Q}_{p})$ on the field $\mathbb{Q}_{p}$ of the $p$-adic
rationals with $n$ arbitrary positive integer.
