{
  "id": "2401.13859",
  "version": "v1",
  "published": "2024-01-24T23:51:00.000Z",
  "updated": "2024-01-24T23:51:00.000Z",
  "title": "Proving the 5-Engel identity in the 2-generator group of exponent four",
  "authors": [
    "Colin Ramsay"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "It is known that the fifth Engel word $E_5$ is trivial in the 2-generator group of exponent four $B(2,4)$, and so can be written as a product of fourth powers. Explicit products of 250 and 28 powers are known, using fourth powers of words up to lengths four and ten respectively. Using a reduction technique based on the recursive enumerability of the set of trivial words in a finite presentation we were able to rewrite $E_5$ as a product of 26 fourth powers of words up to length five.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-24T23:51:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fourth powers",
      "fifth engel word",
      "finite presentation",
      "reduction technique",
      "trivial words"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}