scholarly journals RIGHT ENGEL-TYPE SUBGROUPS AND LENGTH PARAMETERS OF FINITE GROUPS

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.

Generalized Fitting subgroup of a group of finite Morley rank

1991 ◽  
Vol 56 (4) ◽  
pp. 1391-1399 ◽  
Author(s):  
Ali Nesin
Keyword(s):  
Finite Group ◽  
Morley Rank ◽  
Fitting Subgroup ◽  
Generalized Fitting Subgroup ◽  
Finite Morley Rank ◽  
The Generalized Fitting Subgroup ◽  
A Finite Group ◽  
Subnormal Subgroups

AbstractWe define a characteristic and definable subgroup F*(G) of any group G of finite Morley rank that behaves very much like the generalized Fitting subgroup of a finite group. We also prove that semisimple subnormal subgroups of G are all definable and that there are finitely many of them.

scholarly journals On Finite Groups with an Automorphism of Prime Order Whose Fixed Points Have Bounded Engel Sinks

2021 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Fixed Points ◽  
Finite Groups ◽  
Prime Order ◽  
Fitting Subgroup ◽  
Engel Element ◽  
A Finite Group

AbstractA left Engel sink of an elementgof a groupGis a set$${\mathscr {E}}(g)$$E(g)such that for every$$x\in G$$x∈Gall sufficiently long commutators$$[...[[x,g],g],\dots ,g]$$[...[[x,g],g],⋯,g]belong to$${\mathscr {E}}(g)$$E(g). (Thus,gis a left Engel element precisely when we can choose$${\mathscr {E}}(g)=\{ 1\}$$E(g)={1}.) We prove that if a finite groupGadmits an automorphism$$\varphi $$φof prime order coprime to |G| such that for some positive integermevery element of the centralizer$$C_G(\varphi )$$CG(φ)has a left Engel sink of cardinality at mostm, then the index of the second Fitting subgroup$$F_2(G)$$F2(G)is bounded in terms ofm. A right Engel sink of an elementgof a groupGis a set$${\mathscr {R}}(g)$$R(g)such that for every$$x\in G$$x∈Gall sufficiently long commutators$$[\ldots [[g,x],x],\dots ,x]$$[…[[g,x],x],⋯,x]belong to$${\mathscr {R}}(g)$$R(g). (Thus,gis a right Engel element precisely when we can choose$${\mathscr {R}}(g)=\{ 1\}$$R(g)={1}.) We prove that if a finite groupGadmits an automorphism$$\varphi $$φof prime order coprime to |G| such that for some positive integermevery element of the centralizer$$C_G(\varphi )$$CG(φ)has a right Engel sink of cardinality at mostm, then the index of the Fitting subgroup$$F_1(G)$$F1(G)is bounded in terms ofm.

scholarly journals SCHREIER CONDITIONS ON CHIEF FACTORS AND RESIDUALS OF SOLVABLE-LIKE GROUP FORMATIONS

2008 ◽  
Vol 78 (1) ◽  
pp. 97-106
Author(s):  
GIL KAPLAN ◽  
DAN LEVY
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Solvable Groups ◽  
Fitting Subgroup ◽  
Group Extensions ◽  
Formation Of Finite Groups ◽  
Generalized Fitting Subgroup ◽  
Chief Factors ◽  
The Generalized Fitting Subgroup ◽  
Structural Characterizations

AbstractLet α be a formation of finite groups which is closed under subgroups and group extensions and which contains the formation of solvable groups. Let G be any finite group. We state and prove equivalences between conditions on chief factors of G and structural characterizations of the α-residual and theα-radical of G. We also discuss the connection of our results to the generalized Fitting subgroup of G.

scholarly journals Finite groups admitting fixed point free automorphisms of order pq

1987 ◽  
Vol 30 (1) ◽  
pp. 51-56 ◽  
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.

scholarly journals On cyclic subgroups of finite groups

1982 ◽  
Vol 25 (1) ◽  
pp. 19-20 ◽  
Author(s):  
U. Dempwolff ◽  
S. K. Wong
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Proof ◽  
Cyclic Subgroup ◽  
Finite Subgroup ◽  
Fitting Subgroup ◽  
Cyclic Subgroups ◽  
Nontrivial Subgroup ◽  
A Finite Group

In [3] Laffey has shown that if Z is a cyclic subgroup of a finite subgroup G, then either a nontrivial subgroup of Z is normal in the Fitting subgroup F(G) or there exists a g in G such that Zg∩Z = 1. In this note we offer a simple proof of the following generalisation of that result:Theorem. Let G be a finite group and X and Y cyclic subgroups of G. Then there exists a g in G such that Xg∩Y⊴F(G).

scholarly journals FINITE GROUPS WITH SOME ℨ-PERMUTABLE SUBGROUPS

2009 ◽  
Vol 52 (1) ◽  
pp. 145-150 ◽  
Author(s):  
YANGMING LI ◽  
LIFANG WANG ◽  
YANMING WANG
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Fitting Subgroup ◽  
Sylow Subgroups ◽  
Permutable Subgroups ◽  
Complete Set ◽  
A Finite Group

AbstractLet ℨ be a complete set of Sylow subgroups of a finite group G; that is to say for each prime p dividing the order of G, ℨ contains one and only one Sylow p-subgroup of G. A subgroup H of G is said to be ℨ-permutable in G if H permutes with every member of ℨ. In this paper we characterise the structure of finite groups G with the assumption that (1) all the subgroups of Gp ∈ ℨ are ℨ-permutable in G, for all prime p ∈ π(G), or (2) all the subgroups of Gp ∩ F*(G) are ℨ-permutable in G, for all Gp ∈ ℨ and p ∈ π(G), where F*(G) is the generalised Fitting subgroup of G.

On residuals of finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefanos Aivazidis ◽  
Thomas Müller
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Fitting Subgroup ◽  
Soluble Groups ◽  
A Finite Group ◽  
Fitting Formation ◽  
Original Theorem

Abstract Theorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups Geom. Dyn. 1 (2007), 4, 401–407] asserts that, in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order and that the inequality is sharp. Inspired by this result and some of the arguments in the above article, we establish the following generalisation: if 𝔛 is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X ¯ \overline{\mathfrak{X}} is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical, | G X ¯ | > | G | γ \lvert G^{\overline{\mathfrak{X}}}\rvert>\lvert G\rvert^{\gamma} , where G X ¯ G^{\overline{\mathfrak{X}}} is the X ¯ \overline{\mathfrak{X}} -residual of 𝐺. When X = N \mathfrak{X}=\mathfrak{N} , the class of finite nilpotent groups, it follows that X ¯ = S \overline{\mathfrak{X}}=\mathfrak{S} , the class of finite soluble groups; thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson’s classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations 𝔛 of full characteristic such that S ⊂ X ¯ ⊂ E \mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E} , where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

scholarly journals Disjoint conjugates of cyclic subgroups of finite groups

1977 ◽  
Vol 20 (3) ◽  
pp. 229-232 ◽  
Author(s):  
Thomas J. Laffey
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Affirmative Answer ◽  
Fitting Subgroup ◽  
Cyclic Subgroups ◽  
Result Theorem ◽  
Nontrivial Subgroup ◽  
A Finite Group

In an earlier paper (2) we considered the following question “If S is a cyclic subgroup of a finite group G and S ∩ F(G) = 1, where F(G) is the Fitting subgroup of G, does there necessarily exist a conjugate Sx of S in G with S ∩ Sx = l?” and we gave an affirmative answer for G simple or soluble. In this paper we answer the question affirmatively in general (in fact we prove a somewhat stronger result (Theorem 3)). We give an example of a group G with a cyclic subgroup S such that (i) no nontrivial subgroup of S is normal in G and (ii) no x exists for which S ∩ Sx = 1.

scholarly journals On the commuting probability for subgroups of a finite group

2021 ◽  
pp. 1-14
Author(s):  
Eloisa Detomi ◽  
Pavel Shumyatsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Commutator Subgroup ◽  
Lower Central Series ◽  
Central Series ◽  
Fitting Subgroup ◽  
Typical Application ◽  
The Generalized Fitting Subgroup ◽  
A Finite Group ◽  
Commuting Probability

Let $K$ be a subgroup of a finite group $G$ . The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$ . Assume that $Pr(K,G)\geq \epsilon$ for some fixed $\epsilon >0$ . We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indices $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$ -bounded. This extends the well-known theorem, due to P. M. Neumann, that covers the case where $K=G$ . We deduce a number of corollaries of this result. A typical application is that if $K$ is the generalized Fitting subgroup $F^{*}(G)$ then $G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$ -bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$ , or a Sylow subgroup, etc.

Groups whose vanishing class sizes are prime powers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sajjad Mahmood Robati ◽  
Roghayeh Hafezieh Balaman
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
The Other ◽  
Fitting Subgroup ◽  
Other Hand ◽  
Special Restriction ◽  
Vanishing Elements ◽  
A Finite Group

Abstract For a finite group 𝐺, an element is called a vanishing element of 𝐺 if it is a zero of an irreducible character of 𝐺; otherwise, it is called a non-vanishing element. Moreover, the conjugacy class of an element is called a vanishing class if that element is a vanishing element. In this paper, we describe finite groups whose vanishing class sizes are all prime powers, and on the other hand we show that non-vanishing elements of such a group lie in the Fitting subgroup which is a proof of a conjecture mentioned in [I. M. Isaacs, G. Navarro and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra 222 (1999), 2, 413–423] under this special restriction on vanishing class sizes.

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