On the generalized Fitting height and insoluble length of finite groups

Robert M. Guralnick; Gareth Tracey

doi:10.1112/blms.12372

On the generalized Fitting height and insoluble length of finite groups

Bulletin of the London Mathematical Society ◽

10.1112/blms.12372 ◽

2020 ◽

Vol 52 (5) ◽

pp. 924-931

Author(s):

Robert M. Guralnick ◽

Gareth Tracey

Keyword(s):

Finite Groups ◽

Fitting Height

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Finite groups admitting fixed point free automorphisms of order pq

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500017958 ◽

1987 ◽

Vol 30 (1) ◽

pp. 51-56 ◽

Cited By ~ 5

Author(s):

Cheng Kai-Nah

Keyword(s):

Fixed Point ◽

Finite Group ◽

Finite Groups ◽

Fitting Height ◽

A Finite Group

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.

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RIGHT ENGEL-TYPE SUBGROUPS AND LENGTH PARAMETERS OF FINITE GROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788719000181 ◽

2019 ◽

Vol 109 (3) ◽

pp. 340-350

Author(s):

E. I. KHUKHRO ◽

P. SHUMYATSKY ◽

G. TRAUSTASON

Keyword(s):

Finite Group ◽

Finite Groups ◽

Fitting Subgroup ◽

Generalized Fitting Subgroup ◽

Fitting Height ◽

Engel Element ◽

The Generalized Fitting Subgroup ◽

Baer’S Theorem ◽

A Finite Group ◽

The Right

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.

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