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A generalization of the sum of element orders function
Abstract
In this paper, for a finite group G and a positive integer k, we introduce the function ψ''k(G)=ψk(G)/(|G|k+1), where ψk(G)=∑g ∈ Go(g)k, as the k-th generalization of the function ψ'', introduced by M. T{ă}rn{ă}uceanu. Different thresholds for cyclic, abelian, nilpotent and (super)solvable groups in terms of k are obtained. It is shown that these thresholds often outperform similar thresholds derived from other parameters like ψ' and ψ''. Moreover, considering ψ''k as a (0,1]-valued function, we study its injectivity, surjectivity and density.
Related
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- The number of distinct element orders in a finite group
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BibTeX
@misc{chatterjee2026generalization,
title = {{A generalization of the sum of element orders function}},
author = {Sayanta Chatterjee and Angsuman Das and Sauvik Poddar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11804},
eprint = {2610.11804},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11804},
}
RIS
TY - UNPB
TI - A generalization of the sum of element orders function
AU - Sayanta Chatterjee
AU - Angsuman Das
AU - Sauvik Poddar
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11804
UR - https://arxiv.org/abs/2610.11804
AB - In this paper, for a finite group G and a positive integer k, we introduce the function ψ''_k(G)=ψ_k(G)/(|G|^k+1), where ψ_k(G)=∑_g ∈ Go(g)^k, as the k-th generalization of the function ψ'', introduced by M. T{ă}rn{ă}uceanu. Different thresholds for cyclic, abelian, nilpotent and (super)solvable groups in terms of k are obtained. It is shown that these thresholds often outperform similar thresholds derived from other parameters like ψ' and ψ''. Moreover, considering ψ''_k as a (0,1]-valued function, we study its injectivity, surjectivity and density.
ER -