Odd square roots and the sum of element orders on Sn and An
Abstract
For a finite group G let ψ(G) be the sum of the orders of its elements. The quotient Sn/An has order two. Hence ψ(Sn/An)=3. The quantity to be compared with ψ(Sn) is therefore 3ψ(An). Computation indicates that 3ψ(An)>ψ(Sn) for every n≥3. We reduce this inequality to a question on the parity of square roots. Let r(β) be the number of odd permutations σ with σ2=β. We show that the sum of the orders of the odd permutations equals 2∑β∈ Anr(β)o(β). We also show that ∑β∈ Anr(β) equals |An|. The inequality then says that r and the order function are negatively correlated on An. It says equally that the average order of an odd permutation is less than twice the average order of an even one. We then rule out two natural approaches. No injection from the odd permutations into An can halve the order at every point. The smallest failure occurs at n=12. A threshold argument through Landau's function fails as well. Computations up to n=60 are reported.
Related
- A non-autonomous Hamiltonian diffeomorphism with roots of all orders.
- Quantitative linear independence for square roots
- Element Orders and Character Codegrees of Finite Groups
- Positive Eigenvalues of Hadamard Square Roots on Finite Abelian Groups
- Finite groups with large sum of element orders compared to cyclic groups
BibTeX
@misc{singh2026square,
title = {{Odd square roots and the sum of element orders on Sn and An}},
author = {Manoj Kumar Singh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05805},
eprint = {2609.05805},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05805},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Odd square roots and the sum of element orders on Sn and An
AU - Manoj Kumar Singh
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05805
UR - https://arxiv.org/abs/2609.05805
AB - For a finite group G let ψ(G) be the sum of the orders of its elements. The quotient S_n/A_n has order two. Hence ψ(S_n/A_n)=3. The quantity to be compared with ψ(S_n) is therefore 3ψ(A_n). Computation indicates that 3ψ(A_n)>ψ(S_n) for every n≥3. We reduce this inequality to a question on the parity of square roots. Let r(β) be the number of odd permutations σ with σ^2=β. We show that the sum of the orders of the odd permutations equals 2∑_β∈ A_nr(β)o(β). We also show that ∑_β∈ A_nr(β) equals |A_n|. The inequality then says that r and the order function are negatively correlated on A_n. It says equally that the average order of an odd permutation is less than twice the average order of an even one. We then rule out two natural approaches. No injection from the odd permutations into A_n can halve the order at every point. The smallest failure occurs at n=12. A threshold argument through Landau's function fails as well. Computations up to n=60 are reported.
ER -