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RIGHT ENGEL-TYPE SUBGROUPS AND LENGTH PARAMETERS OF FINITE GROUPS
| Content Provider | Scilit |
|---|---|
| Author | Khukhro, E. I. Shumyatsky, P. Traustason, G. |
| Copyright Year | 2019 |
| Description | Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,_{n}x]$ over $x\in G$ . In the case when $G$ is soluble we prove that if, for some $n$ , the Fitting height of $R_{n}(g)$ is equal to $k$ , then $g$ belongs to the $(k+1)$ th Fitting subgroup $F_{k+1}(G)$ . For nonsoluble $G$ , it is proved that if, for some $n$ , the generalized Fitting height of $R_{n}(g)$ is equal to $k$ , then $g$ belongs to the generalized Fitting subgroup $F_{f(k,m)}^{\ast }(G)$ with $f(k,m)$ depending only on $k$ and $m$ , where $|g|$ is the product of $m$ primes counting multiplicities. It is also proved that if, for some $n$ , the nonsoluble length of $R_{n}(g)$ is equal to $k$ , then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$ . Earlier, similar generalizations of Baer’s theorem (which states that an Engel element of a finite group belongs to the Fitting subgroup) were obtained by the first two authors in terms of left Engel-type subgroups. |
| Related Links | http://eprints.lincoln.ac.uk/35464/1/khu-shu-tra181.pdf https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F5DF2BA6177FD1EFA57CE6F032F68BD7/S1446788719000181a.pdf/div-class-title-right-engel-type-subgroups-and-length-parameters-of-finite-groups-div.pdf |
| Ending Page | 350 |
| Page Count | 11 |
| Starting Page | 340 |
| ISSN | 14467887 |
| e-ISSN | 14468107 |
| DOI | 10.1017/s1446788719000181 |
| Journal | Journal of the Australian Mathematical Society |
| Issue Number | 3 |
| Volume Number | 109 |
| Language | English |
| Publisher | Cambridge University Press (CUP) |
| Publisher Date | 2020-12-01 |
| Access Restriction | Open |
| Subject Keyword | Journal of the Australian Mathematical Society Fitting Height Nonsoluble Length Generalized Fitting Height Finite Group |
| Content Type | Text |
| Resource Type | Article |
| Subject | Mathematics |
