{
  "id": "2008.08192",
  "version": "v1",
  "published": "2020-08-18T23:25:23.000Z",
  "updated": "2020-08-18T23:25:23.000Z",
  "title": "On the exponent of the Weak commutativity group $χ(G)$",
  "authors": [
    "R. Bastos",
    "E. de Melo",
    "R. de Oliveira"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "The weak commutativity group $\\chi(G)$ is generated by two isomorphic groups $G$ and $G^{\\varphi }$ subject to the relations $[g,g^{\\varphi}]=1$ for all $g \\in G$. The group $\\chi(G)$ is an extension of $D(G) = [G,G^{\\varphi}]$ by $G \\times G$. We prove that if $G$ is a finite solvable group of derived length $d$, then $\\exp(D(G))$ divides $\\exp(G)^{d}$ if $|G|$ is odd and $\\exp(D(G))$ divides $2^{d-1}\\cdot \\exp(G)^{d}$ if $|G|$ is even. Further, if $p$ is a prime and $G$ is a $p$-group of class $p-1$, then $\\exp(D(G))$ divides $\\exp(G)$. Moreover, if $G$ is a finite $p$-group of class $c\\geq 2$, then $\\exp(D(G))$ divides $\\exp(G)^{\\lceil \\log_{p-1}(c+1)\\rceil}$ ($p\\geq 3$) and $\\exp(D(G))$ divides $2^{\\lfloor \\log_2(c)\\rfloor} \\cdot \\exp(G)^{\\lfloor \\log_2(c)\\rfloor+1}$ ($p=2$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-18T23:25:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak commutativity group",
      "isomorphic groups",
      "finite solvable group",
      "derived length"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}