  The power graph of a group $G$, denoted as $P(G)$, constitutes a simple
undirected graph characterized by its vertex set $G$. Specifically, vertices
$a,b$ exhibit adjacency exclusively if $a$ belongs to the cyclic subgroup
generated by $b$ or vice versa. The corresponding proper power graph of $G$ is
obtained by taking $P(G)$ and removing a vertex corresponding to the identity
element, which is denoted as $P^*(G)$. In the context of finite abelian groups,
this article establishes the sufficient and necessary conditions for the proper
power graph's connectedness. Moreover, a precise upper bound for the diameter
of $P^*(G)$ in finite abelian groups is provided with sharpness. This article
also explores the study of vertex connectivity, center, and planarity.
