{
  "id": "2009.09958",
  "version": "v1",
  "published": "2020-09-21T15:33:06.000Z",
  "updated": "2020-09-21T15:33:06.000Z",
  "title": "Embedding theorems for solvable groups",
  "authors": [
    "Vitaly Roman'kov"
  ],
  "comment": "11 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group $G$ lying in a variety ${\\mathcal M}$ can be embedded in a $4$-generated group $H \\in {\\mathcal M}{\\mathcal A}$ (${\\mathcal A}$ means the variety of abelian groups). If $G$ is a finite group, then $H$ can also be found as a finite group. It follows, that any finitely generated (finite) solvable group $G$ of the derived length $l$ can be embedded in a $4$-generated (finite) solvable group $H$ of length $l+1$. Thus, we answer the question of V. H. Mikaelian and A.Yu. Olshanskii. It is also shown that any countable group $G\\in {\\mathcal M}$, such that the abelianization $G_{ab}$ is a free abelian group, is embeddable in a $2$-generated group $H\\in {\\mathcal M}{\\mathcal A}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-21T15:33:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F16",
      "20E22"
    ],
    "keywords": [
      "solvable group",
      "embedding theorems",
      "generated group",
      "finite group",
      "free abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}