On Weakly ℋ-Subgroups of Finite Groups

Let G be a finite group. A subgroup H of G is called an ?-subgroup in G if N _G (H) ∩ H ^x  ≤ H for all x ∈ G. A subgroup H of G is called weakly ?-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an ?-subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly ?-subgroups in G. Some recent results are extended and generalized.     

On Weakly ℋ-embedded Subgroups of Finite Groups

Let G be a finite group and H a subgroup of G. We say that H is an ?-subgroup in G if N_G(H) ∩ H^g ≤ H for all g ∈ G; H is called weakly ?-subgroup in G if G has a normal subgroup K such that G = HK and H ∩ K is an ?-subgroup in G. We say that H is weakly ? -embedded in G if G has a normal subgroup K such that H^G = HK and H ∩ K is an ?-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that some subgroups of prime power order are weakly ?-embedded in G. Our results improve and generalize several recent results in the literature.     

Normal Restriction in Finite Groups

A subgroup H of a finite group G is called an NR ? subgroup (Normal Restriction) if whenever K ? H, then K ^G ∩ H = K, where K ^G is the normal closure of K in G. In this article, we will prove some sufficient conditions for the solvability of finite groups which possess many NR-subgroups. We also prove a criterion for the existence of a normal p-complement in finite groups.     

On Infinite Camina Groups

A group G is called a Camina group if G′ ≠ G and each element x ∈ G?G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).     

Solvability of a Commutative Algebra which Satisfies (x 2)2 = 0

We studied the solvability of the algebra which satisfies the polynomial identity (x ²)² = 0. We believe that, if A is a finite dimensional commutative algebra over a field F of characteristic not 2 which satisfies (x ²)² = 0 for all x ∈ A, then A is solvable. In this article we proved this when dim  _F A ≤ 7.     

Absolute Valued Algebras Satisfying ( x 2, x 2, x 2) = 0

Abstract

Let A be an absolute valued algebra. We prove that if A satisfies the identity (x ², x ², x ²) = 0 for all x in A, and contains a central idempotent e, that is ex = xe for all x in A, then A is finite dimensional. This result enables us to prove that if A satisfies (x ², x ², x ²) = 0 and admits an involution then A is finite dimensional. To show that our assumptions on A are essential we recall that in El-Mallah [El-Mallah, M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51:39–49] it was shown that the existence of a central idempotent in A is not a sufficient condition for A to be finite dimensional; and the example given in El-Mallah [El-Mallah, M. L. (2003). Semi-algebraic absolute valued algebras with an involution. Comm. Algebra 31(7):3135–3141] shows that there exist infinite dimensional semi-algebraic absolute valued algebras satisfying the identity (x ², x ², x ²) = 0.     

On the Genus of the Nilpotent Graphs of Finite Groups

The nilpotent graph of a group G is a simple graph whose vertex set is G?nil(G), where nil(G) = {y ∈ G | ? x, y ? is nilpotent ? x ∈ G}, and two distinct vertices x and y are adjacent if ? x, y ? is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.     

A Note on Projective and Flat Dimensions and the Bieri-Neumann-Strebel-Renz Σ-Invariants

Let G be a finitely generated group, and A a ?[G]-module of flat dimension n such that the homological invariant Σ ⁿ (G, A) is not empty. We show that A has projective dimension n as a ?[G]-module. In particular, if G is a group of homological dimension hd(G) = n such that the homological invariant Σ ⁿ (G, ?) is not empty, then G has cohomological dimension cd(G) = n. We show that if G is a finitely generated soluble group, the converse is true subject to taking a subgroup of finite index, i.e., the equality cd (G) = hd(G) implies that there is a subgroup H of finite index in G such that Σ^∞(H, ?) ≠ ?.     

A New Characterization of A 10 by its Noncommuting Graph

Let G be a nonabelian group and associate a noncommuting graph ?(G) with G as follows: The vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture.

Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ? M, where N(G):={n ∈ ? | G has a conjugacy class of size n}.

In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows.

AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ?(G) ? ? (M). Then G ? M.

In this short article we prove that if G is a finite group with ?(G) ? ? (A ₁₀), then G ? A ₁₀, where A ₁₀ is the alternating group of degree 10.     

On Residually Finite Groups with Engel-like Conditions

Let m, n be positive integers. Suppose that G is a residually finite group in which for every element x ∈ G there exists a positive integer q = q(x) ≤ m such that x^q is left n-Engel. We show that G is locally virtually nilpotent. Further, let w be a multilinear commutator and G a residually finite group in which for every product of at most 896 w-values x there exists a positive integer q = q(x) dividing m such that x^q is left n-Engel. Then w(G) is locally virtually nilpotent.     

On CAP*-subalgebras of Lie Algebras

A subalgebra H of a Lie algebra L is said a CAP*-subalgebra if, for any non-Frattini chief factor A/B of L, we have H + A = H + B or H ∩ A = H ∩ B. In this article, using this concept, we give some characterizations of solvability and supersolvability of a finite dimensional Lie algebra.     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

On Π-Supplemented Subgroups of a Finite Group

A subgroup H of a finite group G is said to satisfy Π-property in G if for every chief factor L/K of G, |G/K: N_G/K(HK/K ∩ L/K)| is a π(HK/K ∩ L/K)-number. A subgroup H of G is called Π-supplemented in G if there exists a subgroup T of G such that G = HT and H ∩ T ≤ I ≤ H, where I satisfies Π-property in G. In this article, we investigate the structure of a finite group G under the assumption that some primary subgroups of G are Π-supplemented in G. The main result we proved improves a large number of earlier results.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

Noncyclic Graph of a Group

We associate a graph Γ _G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ? G| 〈x, y〉 is cyclic for all y ? G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ _G is finite if and only if Γ _G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ _G  ? Γ _H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ _G  ? Γ _H for some group H, then G ? H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ _G  ? Γ _H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.     

Module Correspondences for Twisted Group Algebras

Abstract

Let G and A be finite groups such that (|G|, |A|) = 1. Let K be an algebraically closed field with Char K = 0. Denote by K ^α G the twisted group algebra of G over K with factor set α. In this paper we prove that if A acts homogeneously on K ^α G, then there exists an action of A on G, and there is a one-to-one correspondence between the set of A-invariant irreducible K ^α G-modules and the set of irreducible K ^α C _G (A)-modules.     

Some Properties of Bell Groups

For any integer n ≠ 0,1, a group G is said to be “n-Bell” if it satisfies the identity [x ⁿ ,y] = [x,y ⁿ ]. It is known that if G is an n-Bell group, then the factor group G/Z ₂(G) has finite exponent dividing 12n ⁵(n ? 1)⁵. In this article we show that this bound can be improved. Moreover, we prove that every n-Bell group is n-nilpotent; consequently, using results of Baer on finite n-nilpotent groups, we give the structure of locally finite n-Bell groups. Finally, we are concerned with locally graded n-Bell groups for special values of n.     

Remotal Sets in Orlicz Spaces

Let X be a Banach space, (I, μ) be a finite measure space. By L ^Φ(I, X), let us denote the space of all X-valued Bochner Orlicz integrable functions on the unit interval I equipped with the Luxemburg norm. A closed bounded subset G of X is called remotal if for any x ∈ X, there exists g ∈ G such that ‖x ? g‖ = ρ(x, G) = sup {‖x ? y‖: y ∈ G}. In this article, we show that for a separable remotal set G ? X, the set of Bochner integrable functions, L ^Φ(I, G) is remotal in L ^Φ(I, X). Some other results are presented.     

On Finite p-Groups Whose IA-Automorphisms Are All Class Preserving

Let G be a finite non-abelian p-group, where p is a prime. An automorphism α of G is called a class preserving automorphism if α(x) ∈ x ^G the conjugacy class of x in G, for all x ∈ G. An automorphism α of G is called an IA-automorphism if x ^?1α(x) ∈ G′ for each x ∈ G. In this paper, we give necessary and sufficient conditions on finite p-group G of nilpotency class 2 such that every IA-automorphism is class preserving.     

New Characterizations of p-Supersolubility of Finite Groups

Let H be a subgroup of a finite group G. H is said to be λ-supplemented in G if G has a subgroup T such that G = HT and H ∩ T ≤ H _SE , where H _SE denotes the subgroup of H generated by all those subgroups of H, which are S-quasinormally embedded in G. In this article, some results about the λ-supplemented subgroups are obtained, by which we determine the structure of some classes of finite groups. In particular, some new characterizations of p-supersolubility of finite groups are given under the assumption that some primary subgroups are λ-supplemented. As applications, a number of previous known results are generalized.