On Huppert's ρ-σ conjecture for primes in the orders of elements

Abstract

Let G be a finite group, let ρ(G) denote the number of distinct prime divisors of |G|, and let σ(G) be the maximal number of distinct prime divisors of the order of an element of G. Let ρ(n)=max{ρ(G) | G solvable, σ(G)=n}. We prove that ρ(n)≤ 4n−1+π(2n) for all n, and, with the help of a computer, that ρ(n)≤ 9n/2 for all n≤ 16. In particular, ρ(n)≤⌊ 9n/2⌋+2 and ρ(n)≤ (101/22)n for all n, and ρ(n)≤ 9n/2 for all n≥ 58. This improves the bound ρ(n)≤ (17/3)n due to Yang, and it shows that the bound ρ(n)≤ 5n stated by Bellotti, Keller and Trudgian holds, although its original proof contains a gap. Moreover, we show that ρ(n)≥ 3n for all n≥ 4, which proves one half of a conjecture of the second author stating that ρ(n)=3n for n≥ 4. We also derive a consequence for character codegrees and, conditionally, for arbitrary finite groups. The proofs of the main results do not rely on the classification of finite simple groups.

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Cite

BibTeX RIS

BibTeX

@misc{jabara2026hupperts,
  title = {{On Huppert's $ρ$-$σ$ conjecture for primes in the orders of elements}},
  author = {Enrico Jabara and Thomas Michael Keller},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25571},
  eprint = {2610.25571},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25571},
}

RIS

TY  - UNPB
TI  - On Huppert's ρ-σ conjecture for primes in the orders of elements
AU  - Enrico Jabara
AU  - Thomas Michael Keller
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25571
UR  - https://arxiv.org/abs/2610.25571
AB  - Let G be a finite group, let ρ(G) denote the number of distinct prime divisors of |G|, and let σ(G) be the maximal number of distinct prime divisors of the order of an element of G. Let ρ(n)=max{ρ(G) | G solvable, σ(G)=n}. We prove that ρ(n)≤ 4n−1+π(2n) for all n, and, with the help of a computer, that ρ(n)≤ 9n/2 for all n≤ 16. In particular, ρ(n)≤⌊ 9n/2⌋+2 and ρ(n)≤ (101/22)n for all n, and ρ(n)≤ 9n/2 for all n≥ 58. This improves the bound ρ(n)≤ (17/3)n due to Yang, and it shows that the bound ρ(n)≤ 5n stated by Bellotti, Keller and Trudgian holds, although its original proof contains a gap. Moreover, we show that ρ(n)≥ 3n for all n≥ 4, which proves one half of a conjecture of the second author stating that ρ(n)=3n for n≥ 4. We also derive a consequence for character codegrees and, conditionally, for arbitrary finite groups. The proofs of the main results do not rely on the classification of finite simple groups.
ER  -

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