scholarly journals On the generalized Fitting height and insoluble length of finite groups

2020 ◽  
Vol 52 (5) ◽  
pp. 924-931
Author(s):  
Robert M. Guralnick ◽  
Gareth Tracey
Keyword(s):  
1987 ◽  
Vol 30 (1) ◽  
pp. 51-56 ◽  
Author(s):  
Cheng Kai-Nah

By the results of Rickman [7] and Ralston [6], a finite group G admitting a fixed point free automorphism α of order pq, where p and q are primes, is soluble. If p = q, then |G| is necessarily coprime to |α|, and it follows from Berger [1] that G has Fitting height at most 2, the composition length of <α>. The purpose of this paper is to prove a corresponding result in the case when p≠q.


2019 ◽  
Vol 109 (3) ◽  
pp. 340-350
Author(s):  
E. I. KHUKHRO ◽  
P. SHUMYATSKY ◽  
G. TRAUSTASON

AbstractLet $g$ 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.


Author(s):  
Simon R. Blackburn ◽  
Peter M. Neumann ◽  
Geetha Venkataraman
Keyword(s):  

2009 ◽  
Author(s):  
Tullio Ceccherini-Silberstein ◽  
Fabio Scarabotti ◽  
Filippo Tolli

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