Generalized Fitting Subgroup Research Articles

Characters, commutators and centers of Sylow subgroups

The character table of a finite group G G determines whether | P : P ′ | = p 2 |P:P’|=p^2 and whether | P : Z ( P ) | = p 2 |P:\mathbf {Z}(P)|=p^2 , where P P is a Sylow p p -subgroup of G G . To prove the latter, we give a detailed classification of those groups in terms of the generalized Fitting subgroup.

Products of F∗(G)-subnormal subgroups of finite groups

A subgroup [Formula: see text] of a group [Formula: see text] is called [Formula: see text]-subnormal if it is subnormal in the product with the generalized Fitting subgroup [Formula: see text] of [Formula: see text]. All saturated formations [Formula: see text] that contain every group [Formula: see text] where [Formula: see text] and [Formula: see text] are [Formula: see text]-subnormal [Formula: see text]-subgroups are described. Moreover, if such formation [Formula: see text] is also normally hereditary, then every group [Formula: see text] where [Formula: see text] and [Formula: see text] are [Formula: see text]-subnormal quasi-[Formula: see text]-subgroups is a quasi-[Formula: see text]-group. The connection of above-mentioned formations to the lattice formations, GWP-formations and formations with the Kegel property is found.

On the commuting probability for subgroups of a finite group

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.

Centers of Sylow subgroups and automorphisms

Suppose that p is an odd prime and G is a finite group having no normal non-trivial p'-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. This answers a conjecture of Gross. An easy corollary is that if p is an odd prime and P is a Sylow p-subgroup of G, then the center of P is contained in the generalized Fitting subgroup of G. We give two proofs both requiring the classification of finite simple groups. For p=2, the result fails but Glauberman in 1968 proved that the square of a is inner. This answered a problem of Kourovka posed in 1999.

RIGHT ENGEL-TYPE SUBGROUPS AND LENGTH PARAMETERS OF FINITE GROUPS

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.

Irreducible induction and nilpotent subgroups in finite groups

Suppose that G is a finite group and H is a nilpotent subgroup of G. If a character of H induces an irreducible character of G, then the generalized Fitting subgroup of G is nilpotent.

Groups with bounded centralizer chains and the Borovik–Khukhro conjecture

Abstract Let G be a locally finite group and let F ⁢ ( G ) {F(G)} be the Hirsch–Plotkin radical of G. Let S denote the full inverse image of the generalized Fitting subgroup of G / F ⁢ ( G ) {G/F(G)} in G. Assume that there is a number k such that the length of every nested chain of centralizers in G does not exceed k. The Borovik–Khukhro conjecture states, in particular, that under this assumption, the quotient G / S {G/S} contains an abelian subgroup of finite index bounded in terms of k. We disprove this statement and prove a weak analogue of it.

Engel-type subgroups and length parameters of finite groups

Let $g$ be an element of a finite group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. By Baer's theorem, if $E_n(g)=1$, then $g$ belongs to the Fitting subgroup $F(G)$. We generalize this theorem in terms of certain length parameters of $E_n(g)$. For soluble $G$ we prove that if, for some $n$, the Fitting height of $E_n(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$ the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height $h^*(H)$ of a finite group $H$ is the least number $h$ such that $F^*_h(H)=H$, where $F^*_0(H)=1$, and $F^*_{i+1}(H)$ is the inverse image of the generalized Fitting subgroup $F^*(H/F^*_{i}(H))$. Let $m$ be the number of prime factors of $|g|$ counting multiplicities. It is proved that if, for some $n$, the generalized Fitting height of $E_n(g)$ is equal to $k$, then $g$ belongs to $F^*_{f(k,m)}(G)$, where $f(k,m)$ depends only on $k$ and $m$. The nonsoluble length~$\lambda (H)$ of a finite group~$H$ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if $\lambda (E_n(g))=k$, then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$. We also state conjectures of stronger results independent of $m$ and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.

Length-type parameters of finite groups with almost unipotent automorphisms

Let $\alpha $ be an automorphism of a finite group $G$. For a positive integer $n$, let $E_{G,n}(\alpha )$ be the subgroup generated by all commutators $[...[[x,\alpha ], \alpha ],\dots ,\alpha ]$ in the semidirect product $G\langle\alpha \rangle$ over $x\in G$, where $\alpha $ is repeated $n$ times. By Baer's theorem, if $E_{G,n}(\alpha )=1$, then the commutator subgroup $[G,\alpha ]$ is nilpotent. We generalize this theorem in terms of certain length parameters of $E_{G,n}(\alpha )$. For soluble $G$ we prove that if, for some $n$, the Fitting height of $E_{G,n}(\alpha )$ is equal to $k$, then the Fitting height of $[G,\alpha ]$ is at most $k+1$. For nonsoluble $G$ the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height $h^*(H)$ of a finite group $H$ is the least number $h$ such that $F^*_h(H)=H$, where $F^*_0(H)=1$, and $F^*_{i+1}(H)$ is the inverse image of the generalized Fitting subgroup $F^*(H/F^*_{i}(H))$. Let $m$ be the number of prime factors of the order $|\alpha |$ counting multiplicities. It is proved that if, for some $n$, the generalized Fitting height of $E_{G,n}(\alpha )$ is equal to $k$, then the generalized Fitting height of $[G,\alpha ]$ is bounded in terms of $k$ and $m$. The nonsoluble length~$\lambda (H)$ of a finite group~$H$ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if $\lambda (E_{G,n}(\alpha ))=k$, then the nonsoluble length of $[G,\alpha ]$ is bounded in terms of $k$ and $m$. We also state conjectures of stronger results independent of $m$ and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.

On the length of a finite group and of its 2-generator subgroups

The nonsoluble length $\lambda(G)$ of a finite group $G$ is defined as the minimum number of nonsoluble factors in a normal series of $G$ each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. In the present paper we prove that if $\lambda (J)\leq k$ for every 2-generator subgroup $J$ of $G$, then $\lambda(G)\leq k$. It is conjectured that if $h^*(J)\leq k$ for every 2-generator subgroup $J$, then $h^*(G)\leq k$. We prove that if $h^*(\langle x,x^g\rangle)\leq k$ for all $x,g\in G$ such that $\langle x,x^g\rangle$ is soluble, then $h^*(G)$ is $k$-bounded.

On the length of a finite group and of its 2-generator subgroups

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h*(G) such that F*h(G) = G, where F*1(G) = F*(G) is the generalized Fitting subgroup, and F*i+1(G) is the inverse image of F* (G/F*i(G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, xg〉) ≤ k for all x, g ∈ G such that 〈x, xg〉 is soluble, then h* (G) is k-bounded.

Special blocks of finite groups

We first determine in this paper the structure of the generalized Fitting subgroup F* (G) of the finite groups G all of whose defect groups (of blocks) are conjugate under the automorphism group Aut(G) to either a Sylow p-subgroup or a fixed p-subgroup of G. Then we prove that if a finite group L acts transitively on the set of its proper Sylow p-intersections, then either L/Op(L) has a T.I. Sylow p-subgroup or p = 2 and the normal closure of a Sylow 2-subgroup of L/O2(L) is 2-nilpotent with completely descripted structure. This solves a long-open problem. We also obtain some generalizations of the classic results by Isaacs and Passman on half-transitivity.

Finite groups all of whose minimal subgroups are N E ∗-subgroups

Let G be a finite group. A subgroup H of G is called an NE-subgroup of G if it satisfies H G ∩N G (H)=H. A subgroup H of G is said to be a N E ∗-subgroup of G if there exists a subnormal subgroup T of G such that G=H T and H∩T is a NE-subgroup of G. In this article, we investigate the structure of G under the assumption that subgroups of prime order are N E ∗-subgroups of G. The finite groups, all of whose minimal subgroups of the generalized Fitting subgroup are N E ∗-subgroups are classified.

On the length of finite factorized groups

The nonsoluble length \(\lambda (G)\) of a finite group \(G\) is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of non-abelian simple groups. The generalized Fitting height of a finite group \(G\) is the least number \(h=h^*(G)\) such that \(F^*_h(G)=G\), where \(F^*_1(G)=F^*(G)\) is the generalized Fitting subgroup, and \(F^*_{i+1}(G)\) is the inverse image of \(F^*(G/F^*_{i}(G))\). It is proved that if a finite group \(G=AB\) is factorized by two subgroups of coprime orders, then the nonsoluble length of \(G\) is bounded in terms of the generalized Fitting heights of \(A\) and \(B\). It is also proved that if, say, \(B\) is soluble of derived length \(d\), then the generalized Fitting height of \(G\) is bounded in terms of \(d\) and the generalized Fitting height of \(A\).

Semisimple groups acting on semisimple groups

Suppose A and B are finite semisimple groups, i.e. direct products of simple nonabelian groups. Assume A and B are subgroups of a group G such that B is normalized by A and \({A \not\le B}\). It is shown that then the centralizer C AB (B) is not trivial provided P(A) ≥ P(B), where P(X) denotes the smallest degree of a faithful permutation representation of a group X. Hence B cannot be the generalized Fitting subgroup or the socle of its normalizer in G. Further consequences are derived.

Probabilistic generation of finite classical groups in odd characteristic by involutions

An involution in a finite n-dimensional classical group G over a field of odd order q is called (α, β)-balanced if the dimension of its fixed point subspace is between αn and βn. Balanced involutions play an important role in recent constructive recognition algorithms for finite classical groups in odd characteristic. For a given sequence of conjugacy classes of balanced involutions in G, a c-tuple (g1, . . . , gc) is a class-random sequence from 𝒳 if, for each i = 1, . . . , c, gi is a uniformly distributed random element of , and the gi are mutually independent. We show that there is a number c = c(α, β) such that for large enough n, for a given such sequence 𝒳 of length c, a class-random sequence from 𝒳 generates a subgroup containing the generalized Fitting subgroup of G with probability at least 1 – q–n.

Character separation and principal covering

We investigate the separation of irreducible characters by blocks at different primes and the covering of irreducible characters by blocks (viewed as sets of characters); these notions are used to prove results on the group structure. The covering of all characters of a group by principal blocks is only possible when already one principal block suffices or the generalized Fitting subgroup has a very special structure.

A characterization of the hypercyclically embedded subgroups of finite groups

A normal subgroup H of a finite group G is said to be hypercyclically embedded in G if every chief factor of G below H is cyclic. The major aim of the present paper is to characterize the normal hypercyclically embedded subgroups E of a group G by means of the embedding of the maximal and minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of E .

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let H sG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H ⩽ H sG and HT = C. Our main result is the following A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgroup F*(G) of G, at least one of the following holds:

On SS-Quasinormal Subgroups of Finite Groups

A subgroup of a group G is said to be Sylow-quasinormal (S-quasinormal) in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be Supplement-Sylow-quasinormal (SS-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this article, we investigate the influence of SS-quasinormal of maximal or minimal subgroups of Sylow subgroups of the generalized Fitting subgroup of a finite group.