{
  "id": "2310.09459",
  "version": "v1",
  "published": "2023-10-14T00:50:05.000Z",
  "updated": "2023-10-14T00:50:05.000Z",
  "title": "A new lower bound for the number of conjugacy classes",
  "authors": [
    "Burcu Çınarcı",
    "Thomas Michael Keller"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "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 $p$-solvable groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-14T00:50:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower bound",
      "conjugacy classes",
      "finite group",
      "solvable groups",
      "stronger conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}