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

2024/02/20 by Mohsen Amiri, Amiri, Mohsen · 1 citation
Engineering · Materials Science · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Quasicrystal Structures and Properties #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2402.13247

openalex publication_date 2024/02/20 · openalex created_date 2024/02/23 · openalex updated_date 2026/07/28

Abstract

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_(n)/(p)× 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.

Cited by

• On the Sum of Element Orders in Finite Abelian Groups

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