A new lower bound for the number of conjugacy classes
| Autor: | Çınarcı, Burcu, Keller, Thomas Michael |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In 2003, H\'{e}thelyi and K\"{u}lshammer proposed that if $G$ is a finite group and $p$ is a prime dividing the group order, then $k(G)\geq 2\sqrt{p-1}$, and they proved this conjecture for solvable $G$ and showed that it is sharp for those primes $p$ for which $\sqrt{p-1}$ is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes $p$, and we prove it for solvable groups, and when $p$ is large, also for arbitrary groups. Comment: A hypothesis for Theorem E is added that was inadvertently omitted in the previous version. Some other small changes have been made |
| Databáze: | arXiv |
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