On a bijection between a finite group to a non-cyclic group with divisibility of element orders

Consider a finite group $G$ of order $n$ with a prime divisor $p$. In this article, we establish, among other results, that if the Sylow $p$-subgroup of $G$ is neither cyclic nor generalized quaternion, then there exists a bijection $f$ from $G$ onto the abelian group $C_{\frac{n}{p}}\times C_p$ such that for every element $x$ in $G$, the order of $x$ divides the order of $f(x)$. This resolves Question 1.5 posed in [15]. As application of our results, we show that the group with the third largest value of the sum of element orders in the set of all finite groups of order $n$ is a solvable $p$-nilpotent group where $p$ is the smallest prime divisor of $n$ such that the Sylow $p$-subgroups are not cyclic.