{ "id": "1401.0177", "version": "v1", "published": "2013-12-31T16:49:27.000Z", "updated": "2013-12-31T16:49:27.000Z", "title": "Automorphism tower problem and semigroup of endomorphisms for free Burnside groups", "authors": [ "Varujan Atabekyan" ], "comment": "5 pages", "journal": "Int. J. Algebra Comput., 23, 1485 (2013)", "doi": "10.1142/S0218196713500318", "categories": [ "math.GR" ], "abstract": "We have proved that the group of all inner automorphisms of the free Burnside group $B(m,n)$ is the unique normal subgroup in $Aut(B(m,n))$ among all its subgroups, which are isomorphic to free Burnside group $B(s,n)$ of some rank $s$ for all odd $n\\ge1003$ and $m>1$. It follows that the group of automorphisms $Aut(B(m,n))$ of the free Burnside group $B(m,n)$ is complete for odd $n\\ge1003$, that is it has a trivial center and any automorphism of $Aut(B(m,n))$ is inner. Thus, for groups $B(m,n)$ is solved the automorphism tower problem and is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, proved that every automorphism of $End(B(m,n))$ is a conjugation by an element of $Aut(B(m,n))$.", "revisions": [ { "version": "v1", "updated": "2013-12-31T16:49:27.000Z" } ], "analyses": { "subjects": [ "20F50", "20F28", "20E36", "20D45", "20B27" ], "keywords": [ "free burnside group", "automorphism tower problem", "endomorphisms", "unique normal subgroup", "inner automorphisms" ], "tags": [ "journal article" ], "note": { "typesetting": "TeX", "pages": 5, "language": "en", "license": "arXiv", "status": "editable", "adsabs": "2014arXiv1401.0177A" } } }