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RIGHT ENGEL-TYPE SUBGROUPS AND LENGTH PARAMETERS OF FINITE GROUPS

Published online by Cambridge University Press:  18 July 2019

E. I. KHUKHRO*
Affiliation:
Charlotte Scott Research Centre for Algebra, University of Lincoln, Lincoln LN6 7TS, UK
P. SHUMYATSKY
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia, DF 70910-900, Brazil e-mail: pavel@unb.br
G. TRAUSTASON
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK e-mail: G.Traustason@bath.ac.uk
*Corresponding

Abstract

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.

Type
Research Article
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by B. Martin

The first author was supported by the Russian Science Foundation, project no. 14-21-00065, the second by FAPDF, Brazil, and the third by EPSRC.

References

Baer, R., ‘Engelsche Elemente Noetherscher Gruppen’, Math. Ann. 133 (1957), 256270.Google Scholar
Hall, P. and Higman, G., ‘On the p-length of p-soluble groups and reduction theorems for Burnside’s problem’, Proc. Lond. Math. Soc. (3) 6 (1956), 142.Google Scholar
Khukhro, E. I. and Shumyatsky, P., ‘Words and pronilpotent subgroups in profinite groups’, J. Aust. Math. Soc. 97 (2014), 343364.Google Scholar
Khukhro, E. I. and Shumyatsky, P., ‘Engel-type subgroups and length parameters of finite groups’, Israel J. Math. 222 (2017), 599629.Google Scholar
Wilson, J. S., ‘On the structure of compact torsion groups’, Monatsh. Math. 96 (1983), 5766.Google Scholar
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