Engel-type subgroups and length parameters of finite groups

Khukhro, Evgeny and Shumyatsky, Pavel (2017) Engel-type subgroups and length parameters of finite groups. Israel Journal of Mathematics, 222 (2). pp. 599-629. ISSN 0021-2172

Full content URL: https://doi.org/10.1007/s11856-017-1601-0

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Engel-type subgroups and length parameters of finite groups
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Abstract

Let g be an element of a finite group G and let Rn(g) be the subgroup generated by all the right Engel values [g,nx] over x∈G. In the case when G is soluble we prove that if, for some n, the Fitting height of Rn(g) is equal to k, then g belongs to the (k+1)th Fitting subgroup Fk+1(G). For nonsoluble G, it is proved that if, for some n, the generalized Fitting height of Rn(g) is equal to k, then g belongs to the generalized Fitting subgroup F∗f(k,m)(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 Rn(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.

Keywords:Finite groups, nonsoluble length, generalized Fitting height, commutator subgroup
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:24937
Deposited On:09 Nov 2016 08:38

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