Finite simple groups in which all maximal subgroups are π-closed. I

Finite simple nonabelian groups G that are not π-closed for some set of primes π but have π-closed maximal subgroups (property (*) for (G, π)) are studied. We give a list L of finite simple groups that contains any group G with the above property (for some π). It is proved that 2 ? π for any pair (G, π) with property (*) (Theorem 1). In addition, we specify for any sporadic simple group G from L all sets of primes π such that the pair (G, π) has property (*) (Theorem 2). The proof uses the author’s results on the control of prime spectra of finite simple groups.     

Essential dimension of algebraic groups,including bad characteristic

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3.     

On the <Emphasis Type="Italic">p</Emphasis>-supersolubility of a finite group with a µ-supplemented sylow <Emphasis Type="Italic">p</Emphasis>-subgroup

A subgroup H of a group G is called µ-supplemented in G if there exists a subgroup K such that G = HK and H ₁ K is a proper subgroup in G for every maximal subgroup H ₁ in H. For the initial values of p, we establish the p-supersolubility of a finite group with a μ-supplemented Sylow p-subgroup.     

A condition for the solvability of finite groups

A subgroup H is called ?-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H ₁ B is a proper subgroup of G for every maximal subgroup H ₁ of H. We investigate the influence of ?-supplementation of Sylow subgroups and obtain a condition for solvability and p-supersolvability of finite groups.     

<Emphasis Type="Italic">c</Emphasis>-Semipermutable subgroups of finite groups

A subgroup is called c-semipermutable in G if A has a minimal supplement T in G such that for every subgroup T ₁ of T there is an element x ∈ T satisfying AT ₁ ^x = T ₁ ^x A. We obtain a few results about the c-semipermutable subgroups and use them to determine the structures of some finite groups.     

Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

Let P be a subgroup of a Sylow subgroup of a finite group G. If P is a Sylow subgroup of some normal subgroup of G then P is called normally embedded in G. We establish tests for a finite group G to be p-supersoluble provided that every maximal subgroup of a Sylow p-subgroup of X is normally embedded in G. We study the cases when X is a normal subgroup of G, X = O_p',p(H), and X = F*(H) where H is a normal subgroup of G.     

On an open problem of Guo-Skiba

Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/A _G of G is called a G-boundary factor of A. For any Gboundary factor H/A _G of A, the subgroup (A ∩ H)/A _G of G/ A _G is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p′-group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups.     

Thompson’s conjecture for the alternating group of degree 2<Emphasis Type="Italic">p</Emphasis> and 2<Emphasis Type="Italic">p</Emphasis>+1

For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ? L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A _i ) is necessarily isomorphic to A _i , where i ∈ {2p, 2p + 1}.     

Products of F*(<Emphasis Type="Italic">G</Emphasis>)-subnormal subgroups of finite groups

A subgroup H of a finite group G is called F*(G)-subnormal if H is subnormal in HF*(G). We show that if a group Gis a product of two F*(G)-subnormal quasinilpotent subgroups, then G is quasinilpotent. We also study groups G = AB, where A is a nilpotent F*(G)-subnormal subgroup and B is a F*(G)-subnormal supersoluble subgroup. Particularly, we show that such groups G are soluble.     

On finite groups for which the lattice of <Emphasis Type="Italic">S</Emphasis>-permutable subgroups is distributive

Let p be a prime and let P be a Sylow p-subgroup of a finite nonabelian group G. Let bcl(G) be the size of the largest conjugacy classes of the group G. We show that if p is an odd prime but not a Mersenne prime or if P does not involve a section isomorphic to the wreath product \({Z_p \wr Z_p}\), then \({|P/O_p(G)| \leq bcl(G)}\).     

On $$\mathcal {}$$-subgroups of finite groups

Let G be a finite group. A subgroup H of G is s-permutable in G if H permutes with every Sylow subgroup of G. A subgroup H of G is called an \(\mathcal {SSH}\)-subgroup in G if G has an s-permutable subgroup K such that \(H^{sG} = HK\) and \(H^g \cap N_K (H) \leqslant H\), for all \(g \in G\), where \(H^{sG}\) is the intersection of all s-permutable subgroups of G containing H. We study the structure of finite groups under the assumption that the maximal or the minimal subgroups of Sylow subgroups of some normal subgroups of G are \(\mathcal {SSH}\)-subgroups in G. Several recent results from the literature are improved and generalized.     

On <Emphasis Type="Italic">G</Emphasis>-covering subgroup systems of finite groups

Let \(\mathcal{F}\) be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for  \(\mathcal{F}\) if \(G\in \mathcal{F}\) whenever \(\Sigma \subseteq \mathcal{F}\). Let p be any prime dividing |G| and P a Sylow p-subgroup of G. Then we write Σ _p to denote the set of subgroups of G which contains at least one supplement to G of each maximal subgroup of P. We prove that the sets Σ _p and Σ _p ∪Σ _q , where q≠p, are G-covering subgroup systems for many classes of finite groups.     

A solvability criterion for finite groups

We show that if for every prime p, the normalizer of a Sylow p-subgroup of a finite group G admits a p-solvable supplement, then G is solvable. This generalizes a solvability criterion of Hall which asserts that a finite group G is solvable if and only if G has a Hall p′-subgroup for every prime p.     

A note on a result of Skiba

A subgroup H of a group G is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ H _sG , where H _sG is the maximal s-permutable subgroup of G contained in H. We improve a nice result of Skiba to get the following

Theorem. Let ? be a saturated formation containing the class of all supersoluble groups

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and let G be a group with E a normal subgroup of G such that G/E ∈ ?. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| are weakly s-permutable in G for all p ∈ π(F*(E)); moreover, we suppose that every cyclic subgroup of P of order 4 is weakly s-permutable in G if P is a nonabelian 2-group and |D| = 2. Then G ∈ ?.

     

On <Emphasis Type="Italic">c</Emphasis><Superscript>#</Superscript>-normal subgroups infinite groups

A subgroup H of a finite group G is called a c^#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩ K is a CAP-subgroup of G: In this paper, we investigate the influence of fewer c^#-normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.     

The embedding of CAP-subgroups in finite groups

This paper features new and unpublished results about the cover-avoidance property presented in the context of previously established ones. The goal of this work is to contribute to the breadth of the current body of knowledge and to fill in some gaps in the literature. On one hand, we summarize and investigate further the relationships between the cover-avoidance property and normal, conjugate, Sylow, and maximal subgroups. On the other hand, we define a subgroup U to be a subCAP-subgroup of the finite group G if there exists a chain of subgroups \({U = U_{0}\leq U_{1}\leq \dotsm \leq U_{r} = G}\) such that U _i-1 is a CAP-subgroup of U _i for all \({i \in \{1, \dots, r\}}\), and show how this property can be used to characterize solvable groups.     

Finite groups with <Emphasis Type="Italic">C</Emphasis>-quasinormal subgroups

Consider some finite group G and a finite subgroup H of G. Say that H is c-quasinormal in G if G has a quasinormal subgroup T such that HT = G and T ∩ H is quasinormal in G. Given a noncyclic Sylow subgroup P of G, we fix some subgroup D such that 1 < |D| < | P| and study the structure of G under the assumption that all subgroups H of P of the same order as D, having no supersolvable supplement in G, are c-quasinormal in G.     

The complete reducibility of some <Emphasis Type="Italic">GF</Emphasis>(2)<Emphasis Type="Italic">A</Emphasis><Subscript>7</Subscript>-modules

It is proved that, if G is a finite group with a nontrivial normal 2-subgroup Q such that G/Q ～= A ₇ and an element of order 5 from G acts freely on Q, then the extension G over Q is splittable, Q is an elementary abelian group, and Q is the direct product of minimal normal subgroups of G each of which is isomorphic, as a G/Q-module, to one of the two 4-dimensional irreducible GF(2)A ₇-modules that are conjugate with respect to an outer automorphism of the group A ₇.     

A criterion for the <Emphasis Type="Italic">F</Emphasis> <Subscript> <Emphasis Type="Italic">π</Emphasis> </Subscript>-residuality of free products with amalgamated cyclic subgroup of nilpotent groups of finite ranks

Let G be the free product of nilpotent groups A and B of finite rank with amalgamated cyclic subgroup H, H ≠ A and H ≠ B. Suppose that, for some set π of primes, the groups A and B are residually F_π, where F_π is the class of all finite p-groups. We prove that G is residually F_π if and only if H is F_π-separable in A and B.     

On a Fitting length conjecture without the coprimeness condition

Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order p ⁿ q, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.