A characterization of alternating groups. II

Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every two non-commuting members of X generate a subgroup isomorphic to A₄ or to A₅. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they generate a subgroup isomorphic to A₄. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X) and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A₅ then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2) The subgroup 〈X^G〉 is a direct product of subgroups 〈C _α〉-generated by some connected components C _α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup isomorphic to A₅. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A₅ or is cyclic of order 5. Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006.     

Algebraic Independence Results Related to 〈 q , r 〉-Number Systems

We define 〈q, r〉-linear arithmetical functions attached to the 〈q, r〉-number systems and give a necessary and sufficient condition for their generating power series to be algebraically independent over \Bbb C(z){\Bbb C}(z) . We also deduce algebraic independence of the functions values at a nonzero algebraic number in the circle of convergence.     

Algebraic Independence Results Related to 〈q, r〉-Number Systems

We define 〈q, r〉-linear arithmetical functions attached to the 〈q, r〉-number systems and give a necessary and sufficient condition for their generating power series to be algebraically independent over . We also deduce algebraic independence of the functions values at a nonzero algebraic number in the circle of convergence.     

Vertices contained in all minimum paired-dominating sets of a tree

A set S of vertices in a graph G is called a paired-dominating set if it dominates V and 〈S〉 contains at least one perfect matching. We characterize the set of vertices of a tree that are contained in all minimum paired-dominating sets of the tree.     

On Frobenius groups with noninvariant factor SL 2(3)

We obtain a nonsimplicity criterion of an infinite group containing an infinite class of Frobenius groups L _g = 〈a, g ⁻¹ ag〉 with complement SL ₂(3). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 765–777, June, 2006.     

Projective line over the finite quotient ring GF(2)[x]/〈x 3 ™ x〉 and quantum entanglement: The Mermin “magic” square/pentagram

In 1993, Mermin gave surprisingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight respectively using what has since been called the Mermin-Peres “magic” square and the Mermin pentagram. The former is a 3×3 array of nine observables commuting pairwise in each row and column and arranged such that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by a similar contradiction. We establish a one-to-one correspondence between the operators of the Mermin-Peres square and the points of the projective line over the product ring GF(2) ⊗ GF(2). Under this map, the concept mutually commuting transforms into mutually distant, and the distinguishing character of the third column’s observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are either both zero divisors or both units. The ten operators of the Mermin pentagram correspond to a specific subset of points of the line over GF(2)[x]/〈x³ ™ x〉. But the situation in this case is more intricate because there are two different configurations that seem to serve our purpose equally well. The first one comprises the three distinguished points of the (sub)line over GF(2), their three “Jacobson” counterparts, and the four points whose both coordinates are zero divisors. The other con.guration features the neighborhood of the point (1, 0) (or, equivalently, that of (0, 1)). We also mention some other ring lines that might be relevant to BKS proofs in higher dimensions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 219–227, May, 2007.     

Dominions of universal algebras and projective properties

Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ∈ A such that every pair of homomorphisms f, g: A → ∈ {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group 〈H, a〉 generated by one element modulo H. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 541–557, September–October, 2008.     

Finite p-groups with automorphism of a special form

Research on finite solvable groups with C-closed invariant subgroups has given rise to groups structured as follows. Let p, q₁, q₂, ..., q_m be distinct primes, n_i be the exponent of p modulo q_i, and n be the exponent of p modulo . Then G = Pλ〈x〉, where P is a group and ; Z_i; here, Z_i and P/Z(P) are elementary Abelian groups of respective orders and pⁿ, |x| = r, the element x acts irreducibly on P/Z(P) and on each of the subgroups Z_i, and . We state necessary and sufficient conditions for such groups to exist. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 379–389, July–August, 2006.     

An extension of Ezeilo's result

Summary In a recent paper[1] Ezeilo considered the nonlinear third order differential equation x‴ + ω(x′)x″ + ω(x)x′ + ϑ(x, x′, x″)=p(t). He proved the ultimate boundedness of the solutions on rather general conditions for the nonlinear terms ϕ, ϕ, ϑ. These conditions (in a little weaker form) are also sufficient in order to prove the existence of forced oscillations in the case when the excitation is ω-periodic. For this purpose the Lerag-Schauder principle in a form suggested by G. Güssefeldt[2] is applicable. Dedicated to ProfessorKarl Klotter on his 70th birthday Entrata in Redazione il 21 ottobre 1971.     

Clan acts and codimension

Let (T, X) be a continuum act, let cd X=n and suppose A is a T-ideal (i.e., a T-invariant subspace of X), such that Hⁿ(A)≠0. We prove that A is a minimal T-ideal iff A=Gx for some x∈X and maximal group G in the minimal ideal of T. Moreover, if these conditions are satisfied, then A is the only minimal T-ideal and also is the unique floor for every nonzero element of Hⁿ(X). We need and also prove here an improved version of the Tube Theorem [3], and this corollary: if (G, X) is an intransitive transformation group with G compact, X locally compact and finite dimensional, and X/G connected, then dimension Gx<dimension X for all x∈X. NSF GP 9659. NSF GP 28655.     

Ornstein-Zernike theory for finite range Ising models above T c

 We derive a precise Ornstein-Zernike asymptotic formula for the decay of the two-point function 〈Σ₀Σ _x 〉_β in the general context of finite range Ising type models on ℤ ^d . The proof relies in an essential way on the a-priori knowledge of the strict exponential decay of the two-point function and, by the sharp characterization of phase transition due to Aizenman, Barsky and Fernández, goes through in the whole of the high temperature region β<β _c . As a byproduct we obtain that for every β<β _c , the inverse correlation length ξ_β is an analytic and strictly convex function of direction. Received: 10 January 2002 / Revised version: 19 June 2002 / Published online: 14 November 2002 Partly supported by Italian G. N. A. F. A, EC grant SC1-CT91-0695 and the University of Bologna. Funds for selected research topics. Partly supported by the ISRAEL SCIENCE FOUNDATION founded by The Israel Academy of Science and Humanities. Partly supported by the Swiss National Science Foundation grant #8220-056599. Mathematics Subject Classification (2000): 60F15, 60K15, 60K35, 82B20, 37C30 Key words or phrases: Ising model – Ornstein-Zernike decay of correlations – Ruelle operator – Renormalization – Local limit theorems     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

Periodic solutions of a third order nonlinear differential equation

Summary The differential equation x‴ + ϕ(x′)x″ + ϕ(x)x′ + f(x)=p(t) is considered where the forcing term p is an ω-periodic function of t. In the special cases ϕ(x)=k² respectively ϕ(x′)=a the existence of periodic solutions is proved on the basis of the Lerag-Schauder fixed point technique. The conditions imposed on the nonlinear terms do not include the ultimate boundedness of all solutions. Entrata in Redazione il 18 settembre 1971.     

Finite neofields and 2-transitive groups

Let N be a finite neofield distinct from the Galois field and let G be a group generated by right translations x→x+a of an additive loop of N. We prove that, except for four particular cases, G=S_N or G=A_N holds. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 166–193, March–April, 1997.     

On the asymptotic behavior of irreducibles in block semigroups

Let G be a finite abelian group and \cal B (G) the block semigroup over G . For an irreducible element x∈ \cal B (G) , we study the asymptotic behavior of the function η(x) , which counts the total number of non-associated irreducible factorizations of x∈ \cal B (G) . Set We give characterizations of irreducibles x such that σ(x)=0 or σ(x)=1 , as well as construct examples of monoids where the maximum and minimum possible values of σ(x) are prescribed positive integers. We show for cyclic groups that the global growth of σ(x) is not polynomial, and close by using σ(x) to characterize elementary p -groups. November 20, 1999     

The ergodic infinite measure preserving transformation of boole

G. Boole proved that the transformation φ of the real line, defined by φ(x)=x−1/x, preserves Lebesgue measure. A general method is applied to proving that φ is ergodic. Some further applications of the method are also indicated.     

Hyperidentities of lattices and semilattices

Following W. Taylor we define a hyperidentity ∈ to be formally the same as an identity (e.g.,F(G(x, y, z), G(x, y, z))=G(x, y, z)). However, a varietyV is said to satisfy a hyperidentity ∈, if whenever the operation symbols of ∈ are replaced by any choice of polynomials (appropriate forV) of the same arity as the corresponding operation symbols of ∈, then the resulting identity holds inV in the usual sense. For example, if a varietyV of type <2,2> with operation symbols ∨ and ∧ satisfies the hyperidentity given above, then substituting the polynomial (x∨y)∨z for the symbolG, and the polynomialx∧y forF, we see thatV must in particular satisfy the identity ((x∨y)∨z)∧((x∨y)∨z)=((x∨y)∨z). The set of all hyperidentities satisfied by a varietyV, will be denoted byH(V). We shall letH ^m (V) be the set of all hyperidentities hoiding inV with operation symbols of arity at mostm, andH _n (V) will denote the set of all hyperidentities ofV with at mostn distinct variables. In this paper we shall show that ifV is a nontrivial variety of lattices or the variety of all semilattices, then for any integersm andn, there exists a hyperidentity ∈ such that ∈ holds inV, and ∈ is not a consequence ofH ^m (V)∪H _n (V). From this it is deduced that the hyperidentities ofV are not finitely based, partly soling a problem of Taylor [7, Problem 3]. The research of the author was supported by NSERC of Canada. Presented by W. Taylor.     

Element orders in coverings of symmetric and alternating groups

We prove that if the factor group H=G/N of a finite group G is isomorphic to a symmetric or alternating group of degree m, where m≥5 and N≠1, then G has an element whose order is distinct from any element’s order in H. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 38, No. 3, pp. 296–315, May–June, 1999.