Khukhro, Evgeny and Shumyatsky, Pavel
(2017)
Engeltype subgroups and length parameters of finite groups.
Israel Journal of Mathematics, 222
(2).
pp. 599629.
ISSN 00212172
Full content URL: https://doi.org/10.1007/s1185601716010
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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 Engeltype subgroups.
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