The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Universal theories for rigid soluble groups

A group is said to be p-rigid, where p is a natural number, if it has a normal series of the form G = G ₁ > G ₂ > … > G _p  > G _p+1 = 1, whose quotients G _i /G _i+1 are Abelian and are torsion free when treated as \mathbbZ \mathbb{Z} [G/G _i ]-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing p-rigid groups in the class of p-soluble groups. It is proved that if F is a free p-soluble group, G is an arbitrary p-rigid group, and W is an iterated wreath product of p infinite cyclic groups, then ∀-theories for these groups satisfy the inclusions A(F) ê A(G) ê A(W) \mathcal{A}(F) \supseteq \mathcal{A}(G) \supseteq \mathcal{A}(W) . We construct an ∃-axiom distinguishing among p-rigid groups those that are universally equivalent to W. An arbitrary p-rigid group embeds in a divisible decomposed p-rigid group M = M(α₁,…, α _p ). The latter group factors into a semidirect product of Abelian groups A ₁ A ₂…A _p , in which case every quotient M _i /M _i+1 of its rigid series is isomorphic to A _i and is a divisible module of rank α_i over a ring \mathbbZ \mathbb{Z} [M/M _i ]. We specify a recursive system of axioms distinguishing among M-groups those that are Muniversally equivalent to M. As a consequence, it is stated that the universal theory of M with constants in M is decidable. By contrast, the universal theory of W with constants is undecidable.     

A Family of Nonnormal Cayley Digraphs

We call a Cayley digraph Γ = Cay(G, S) normal for G if G _R , the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p ² (p odd prime). As a result, a family of nonnormal Cayley digraphs is found. Received February 23, 1998, Revised September 25, 1998, Accepted October 27, 1998     

Irreducible algebraic sets over divisible decomposed rigid groups

A soluble group G is said to be rigid if it contains a normal series of the form G = G ₁ > G ₂ > …> G _p > G _p+1 = 1, whose quotients G _i /G _i+1 are Abelian and are torsion-free when treated as right ℤ[G/G _i ]-modules. Free soluble groups are important examples of rigid groups. A rigid group G is divisible if elements of a quotient G _i /G _i+1 are divisible by nonzero elements of a ring ℤ[G/G _i ], or, in other words, G _i /G _i+1 is a vector space over a division ring Q(G/G _i ) of quotients of that ring. A rigid group G is decomposed if it splits into a semidirect product A ₁ A ₂…A _p of Abelian groups A _i ≅ G _i /G _i+1. A decomposed divisible rigid group is uniquely defined by cardinalities α _i of bases of suitable vector spaces A _i , and we denote it by M(α₁,…, α _p ). The concept of a rigid group appeared in [arXiv:0808.2932v1 [math.GR], ], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [11], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable decomposed divisible rigid group M(α₁,…, α _p ). Our present goal is to derive important information directly about algebraic geometry over M(α₁,… α _p ). Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe groups that are universally equivalent over M(α₁,…, α _p ) using the language of equations.     

On automorphisms of finite Abelian <Emphasis Type="Italic">p</Emphasis>-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A _p . For a finite abelian p-group A of type (k ₁, ..., k _n ), simple necessary and sufficient conditions for an n × n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k ₁, k ₂) is analyzed. This work has begin during the visit of the second author to the Faculty of Mathematics and Computer Science, Nicolaus Copernicus University during the period July 31–August 13, 2005. This visit was supported by the Nicolaus Copernicus University and a grant from Cnpq.     

Steinberg triality groups acting on 2 ? (<Emphasis Type="Italic">v,k</Emphasis>, 1) designs

A 2 - (υ, k, 1) design D = (ℙ,ℙ, ℬ) is a system consisting of a finite set ℙ of υ points and a collection ℬ of ℙ-subsets of ℙ, called blocks, such that each 2-subset of ℙ is contained in precisely one block. Let G be an automorphism group of a 2-(υ, k, 1) design. Delandtsheer proved that if G is block-primitive and D is not a projective plane, then G is almost simple, that is, T ⩽ G ⩽ Aut(T), where T is a non-abelian simple group. In this paper, we prove that T is not isomorphic to ³ D ₄(q). This paper is part of a project to classify groups and designs where the group acts primitively on the blocks of the design.     

A variant of Davenport’s constant

Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of ]n[≔ {1, 2,…, n} such that elements of A are incongruent modulo p and non-zero modulo p. Let k ≥ D(G/|A| be any integer where D(G) denotes the well-known Davenport’s constant. In this article, we prove that for any sequence g ₁, g ₂,…, g _k (not necessarily distinct) in G, one can always extract a subsequence with 1 ≤ ℓ ≤ k such that

An extension of a result of ribe

It is proved that for every 1≦p<∞, 1≦q<∞ and for every sequence {p _n}, 1≦p _n<∞,p _n→p, the spaceX=(Σ⊕l _{p n}) _q (resp.U=(Σ⊕L _{p n}(0, 1)) _q ) is uniformly homeomorphic toX⊕l _p (resp.U⊕L _p(0, 1)). This extends Ribe’s result from the casep=1 to generalp<∞ and thus provides examples of uniformly convex, uniformly homeomorphic Banach spaces which are not Lipschitz equivalent.     

Automorphism Groups of 2-Valent Connected Cayley Digraphs on Regular <Emphasis Type="Italic">p</Emphasis>-Groups

 Let X=Cay(G,S) be a 2-valent connected Cayley digraph of a regular p-group G and let G _R be the right regular representation of G. It is proved that if G _R is not normal in Aut(X) then X≅[2K ₁ ] with n>1, Aut(X) ≅Z ₂ wrZ _2n , and either G=Z _2n+1 =<a> and S={a,a ²ⁿ⁺¹ }, or G=Z _2n ×Z ₂ =<a>×<b> and S={a,ab}. Received: May 26, 1999 Final version received: June 19, 2000     

<Emphasis Type="Italic">p</Emphasis>-groups having few almost-rational irreducible characters

We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ _p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p ² and p ² + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p ² + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p ², and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p ³ are exceptional.     

On semisymmetric cubic graphs of order 6p~2

A graph r is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut(Г) acting transitively on its edge set but not on its vertex set. In the case of G. = A, we call r a semisymmetric graph. The aim of this paper is to investigate (G-)semisymmetric graphs of prime degree. We give a group-theoretical construction of such graphs, and give a classification of semisymmetric cubic graphs of order 6p2 for an odd prime p.     

Linear-topological classification of separableL p-spaces associated with von Neumann algebras of type I

We classify, up to a linear-topological isomorphism, all separableL _p-spaces, 1≤p<∞, associated with von Neumann algebras of type I. In particular, anyL _p-space associated with an infinite-dimensional atomic von Neumann algebra is isomorphic tol _p, or toC _p, or to . Further, anyL _p-space,p∈[1,∞),p∈2 associated with an infinite-dimensional von Neumann algebraM of type I is isomorphic to one of the following nine Banach spaces: l_p, L_p, S_P, C_p, S_p ⊕ L_p, L_p(S_p), C_p ⊕ L_p, L_p(C_p), C_p ⊕ L_p(S_p). In the casep=1 all the spaces in this list are pairwise non-isomorphic. Research supported by the Australian Research Council.     

Automorphism groups of nilpotent groups and spaces

A pair of finitely generated, torsion-free nilpotent groups G₁,G₂ is constructed with the properties that G₁ and G₂ are p-isomorphic for all primes p, yet Aut(G₁) and Aut(G₂) are not isomorphic. The example constructed is compared to an analogous example in the homotopy category of simply connected, finite CW-complexes.     

OD-characterization of Almost Simple Groups Related to U3（5）

Let G be a finite group with order ｜G｜=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph （or Gruenberg- Kegel graph） denoted .by г（G） （or GK（G））. This graph is constructed as follows： The vertex set of it is π（G） = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge （and we write pi - pj） if and only if G contains an element of order pipj. The degree deg（pi） of a vertex pj ∈π（G） is the number of edges incident on pi. We define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that ｜H｜ = ｜G｜ and D（H） = D（G）. Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L ：= U3（5） be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.     

p-Saturated formations

We consider here a nonempty formation F locally defined by a system {F(t)} where for some fixed prime numberp, F(p) = F. We call such formationsp-locally defined. It is shown that everyp-locally defined formation F has the property thatG/Φ _p (G ∈ F) implies thatG ∈ F where Φ _p (G) is thep-Frattini subgroup ofG. If F is a formation of solvable groups, then this property is equivalent to F having ap-local definition. For solvable groups equivalent conditions are stated in terms of the F-projectors, the F-normalizers, and the F-hypercenter.     

On r-recognition by prime graph of Bp(3) where p is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G @ B_p(3){G\cong B_p(3)} or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then G @ B₃(3), C₃(3), D₄(3){G\cong B_3(3), C_3(3), D_4(3)}, or G/O₂(G) @ Aut(²B₂(8)){G/O_2(G)\cong {\rm Aut}(^2B_2(8))}. As a corollary, the main result of the above paper is obtained.     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

A simple map with no prime factors

An ergodic measure-preserving transformationT of a probability space is said to be simple (of order 2) if every ergodic joining λ ofT with itself is eitherμ×μ or an off-diagonal measureμ _S , i.e.,μ _S (A×B)=μ(A∩S ^;−n ;B) for some invertible, measure preservingS commuting withT. Veech proved that ifT is simple thenT is a group extension of any of its non-trivial factors. Here we construct an example of a weakly mixing simpleT which has no prime factors. This is achieved by constructing an action of the countable Abelian group ℤ⊕G, whereG=⊕ _i=1 ^∞ ℤ₂, such that the ℤ-subaction is simple and has centralizer coinciding with the full ℤ⊕G-action.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Upper Bounds for the Laplacian Graph Eigenvalues

We first apply non-negative matrix theory to the matrix K = D A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1 (G) of G and the spectral radius p(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extremal graphs which achieve the upper bounds.