The number of distinct element orders in a finite group
Abstract
For a finite group G let ω(G) denote its spectrum, the set of orders of its elements, and set η(G)=|ω(G)|, the number of distinct element orders. The set ω(G) has been studied intensively, but its cardinality has received little systematic attention as an invariant in its own right. We develop the theory of η. After describing its behaviour under subgroups, quotients, sections, direct products and Frobenius extensions, we prove the two-sided bound 1+∑p||G|vp(exp G)≤η(G)≤τ(exp G), in which the lower equality characterises the CP-groups, those all of whose elements have prime-power order, and the upper equality the groups whose spectrum realises every divisor of the exponent; nilpotent groups lie at the upper end, giving η(G)=τ(exp G). We also prove η(G)≤τ(|G|) with equality only for cyclic groups, and η(G)≥π(|G|)+1 with equality only when every nontrivial element has prime order. We show that η(G)≤3 forces solvability by an argument resting only on Burnside's theorem, and, combining this with the recognition of A5 by its spectrum, that A5 is the unique non-solvable group with η(G)=4. For the symmetric and alternating groups we record self-contained proofs of closed criteria for membership in the spectrum through a single arithmetic function, placing the associated counting sequences in the present framework, and we determine the extreme values of η over all groups of a fixed order.
Related
BibTeX
@misc{singh2026number,
title = {{The number of distinct element orders in a finite group}},
author = {Manoj Kumar Singh and Sumant Kumar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13016},
eprint = {2609.13016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13016},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - The number of distinct element orders in a finite group
AU - Manoj Kumar Singh
AU - Sumant Kumar
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13016
UR - https://arxiv.org/abs/2609.13016
AB - For a finite group G let ω(G) denote its spectrum, the set of orders of its elements, and set η(G)=|ω(G)|, the number of distinct element orders. The set ω(G) has been studied intensively, but its cardinality has received little systematic attention as an invariant in its own right. We develop the theory of η. After describing its behaviour under subgroups, quotients, sections, direct products and Frobenius extensions, we prove the two-sided bound 1+∑_p||G|v_p(exp G)≤η(G)≤τ(exp G), in which the lower equality characterises the CP-groups, those all of whose elements have prime-power order, and the upper equality the groups whose spectrum realises every divisor of the exponent; nilpotent groups lie at the upper end, giving η(G)=τ(exp G). We also prove η(G)≤τ(|G|) with equality only for cyclic groups, and η(G)≥π(|G|)+1 with equality only when every nontrivial element has prime order. We show that η(G)≤3 forces solvability by an argument resting only on Burnside's theorem, and, combining this with the recognition of A_5 by its spectrum, that A_5 is the unique non-solvable group with η(G)=4. For the symmetric and alternating groups we record self-contained proofs of closed criteria for membership in the spectrum through a single arithmetic function, placing the associated counting sequences in the present framework, and we determine the extreme values of η over all groups of a fixed order.
ER -