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 $C_{\frac{n}{p}}\times C_p$ such that for each x∈G, the order of x divides the order of f(x).

KEYWORDS:

• Bijection
• finite group
• order element
• quaternion group

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D60

Acknowledgments

The authors would like to thank the anonymous referee for his/her very useful and valuable comments and suggestions which substantially shortened the proofs in the paper. In particular the referee has corrected the proof of Corollary 3.4(2) which was wrong in the first version. This research was in part supported by a grant from the university of Zanjan.