Abstract
For \(n\in \mathbb {N}\), we denote by \(\pi (n)\) the set of prime divisors of n. For a block B of a finite group G, let \({{\,\mathrm{Irr}\,}}(B)\) be the set of irreducible complex characters of G belonging to B. Let \(\rho (B)\) be the set of those primes dividing the degree of some character in \({{\,\mathrm{Irr}\,}}(B)\), and let \(\sigma (B)\) be the maximal number of primes dividing such a degree. For a solvable group G, we prove that \(|\rho (B)|\le 3\sigma (B)+1\). This provides a block result in the spirit of Huppert’s \(\rho \)-\(\sigma \) conjecture.
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Acknowledgements
The project is supported by NSFC (Grant Nos. 11631011, 11701421, 11871011, and 11871292) and the Science & Technology Development Fund of Tianjin Education Commission for Higher Education (2020KJ010). The first author is grateful to the Beijing International Center for Mathematical Research at Peking University for its support and hospitality. The authors are grateful to the referee for the valuable suggestions and comments.
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Bessenrodt, C., Liu, Y., Lu, Z. et al. On Huppert’s \(\rho \)-\(\sigma \) conjecture for blocks. Arch. Math. 118, 339–347 (2022). https://doi.org/10.1007/s00013-021-01696-9
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DOI: https://doi.org/10.1007/s00013-021-01696-9