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 Cn/p× Cₚ 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 [12].