{
  "id": "2104.03211",
  "version": "v1",
  "published": "2021-04-07T16:01:03.000Z",
  "updated": "2021-04-07T16:01:03.000Z",
  "title": "On the ranks of the additive and the multiplicative groups of a brace",
  "authors": [
    "Andrea Caranti",
    "Ilaria Del Corso"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "In \\cite[Theorem 2.5]{Bac16} Bachiller proved that if $(G, \\cdot, \\circ)$ is a brace of order the power of a prime $p$ and the rank of $(G,\\cdot)$ is smaller than $p-1$, then the order of any element is the same in the additive and multiplicative group. This means that in this case the isomorphism type of $(G,\\circ)$ determines the isomorphism type of $(G,\\cdot)$. In this paper we complement Bachiller's result in two directions. In Theorem 2.2 we prove that if $(G, \\cdot, \\circ)$ is a brace of order the power of a prime $p$, then $(G,\\cdot)$ has small rank (i.e. $< p-1$) if and only if $(G,\\circ)$ has small rank. We also provide examples of groups of rank $p-1$ in which elements of arbitrarily large order in the additive group become of prime order in the multiplicative group. When the rank is larger, orders may increase.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-07T16:01:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D15",
      "20B35",
      "12F10",
      "16T05",
      "20E18"
    ],
    "keywords": [
      "multiplicative group",
      "isomorphism type",
      "small rank",
      "complement bachillers result",
      "arbitrarily large order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}