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Characterization of finite groups by a bijection with a divisible property on the element orders

ABSTRACT Let G be a finite solvable group of order n and p be a prime divisor of n. In this article, we prove 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 such that for each x∈G, the order of x divides the order of f(x).