On difference between the spectra of the simple groups Bn(q) and Cn(q)

We prove that the nonisomorphic simple groups B _n (q) and C _n (q) have different sets of element orders. Original Russian Text Copyright ? 2007 Grechkoseeva M. A. __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 1, pp. 89–92, January–February, 2007.     

The young diagrams of a pair of irreducible characters of S n with the same zero set on S n ?

In studying the pairs of irreducible characters of the symmetric group S _n with the same zero set on A _n or S _n ∖ A _n (as well as the pairs of irreducible characters with the same zero set on the alternating group A _n ), the results are important on the connection between the Young diagrams of the characters of these pairs. We prove a theorem that considerably generalizes two previous results of frequent use in this direction. Original Russian Text Copyright ? 2008 Belonogov V. A. The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00148) and the RFBR-NSFC (Grant 05-01-39000). __________ Ekaterinburg. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 5, pp. 992–1006, September–October, 2008.     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ))

This paper considers the isometric extension problem concerning the mapping from the unit sphere S ₁(E) of the normed space E into the unit sphere S ₁(l ^∞(Γ)). We find a condition under which an isometry from S ₁(E) into S ₁(l ^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V ₀: S ₁(E) → S ₁(F) is a surjective isometry, where c ₀₀(Γ) ⊆ F ⊆ l ^∞(Γ), then V ₀ can be extended to be an isometric operator defined on the whole space. This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614)     

The kernel of the Rost invariant, Serre''s Conjecture II and the Hasse principle for quasi-split groups 3,6 D 4 , E 6 , E 7

 We prove that for a simple simply connected quasi-split group of type ^3,6 D ₄ ,E ₆ ,E ₇ defined over a perfect field F of characteristic ≠=2,3 the Rost invariant has trivial kernel. In certain cases we give a formula for the Rost invariant. It follows immediately from the result above that if cd F≤2 (resp. vcd F≤2) then Serre's Conjecture II (resp. the Hasse principle) holds for such a group. For a (C ₂ )-field, in particular ℂ(x,y), we prove the stronger result that Serre's Conjecture II holds for all (not necessary quasi-split) exceptional groups of type ^3,6 D ₄ ,E ₆ ,E ₇ . Received: 27 March 2002 / Published online: 28 March 2003 The author gratefully acknowledge the support of TMR ERB FMRX CT-97-0107 and Forschungsinstitut für Mathematik, ETH in Zürich     

Best Simultaneous Approximation in L(I,E)

Let G be a reflexive subspace of the Banach space E and let L^p(I,E) denote the space of all p-Bochner integrable functions on the interval I=[0,1] with values in E, 1p∞. Given any norm N( , ) on R², N nondecreasing in each coordinate on the set R²₊, we prove that L^p(I,G) is N-simultaneously proximinal in L^p(I,E). Other results are also obtained.     

Irreducible characters of the group S n that are semiproportional on A n

Previously, we dubbed the conjecture that the alternating group A_n has no semiproportional irreducible characters for any natural n [1]. This conjecture was then shown to be equivalent to the following [3]. Let α and β be partitions of a number n such that their corresponding characters χ^α and χ^β in the group S_n are semiproportional on A_n. Then one of the partitions α or β is self-associated. Here, we describe all pairs (α, β) of partitions satisfying the hypothesis and the conclusion of the latter conjecture. Supported by RFBR (grant No. 07-01-00148) and by RFBR-NSFC (grant No. 05-01-39000). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 135–156, March–April, 2008.     

Expansion in SLd(Z/qZ), q arbitrary

Let S be a fixed finite symmetric subset of SL _d (Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of π _q (G) with respect to the generating set π _q (S) form a family of expanders, where π _q is the projection map Z→Z/q Z.     

Almost simple groups with socle 3D4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O’Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T ≤ G ≤ Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to ³ D ₄(q), then T is line-transitive, where q is a power of the prime p.     

Distributivity conditions for lattices of dominions in quasivarieties of Abelian groups

Let ℳ be any quasivariety of Abelian groups, L_q(ℳ) be a subquasivariety lattice of ℳ, dom _G ^ℳ be the dominion of a subgroup H of a group G in ℳ, and G/dom _G ^ℳ (H) be a finitely generated group. It is known that the set L(G, H, ℳ) = {dom _G ^N (H)| N ∈ L_q(ℳ)} forms a lattice w.r.t. set-theoretic inclusion. We look at the structure of dom _G ^ℳ (H). It is proved that the lattice L(G,H,ℳ) is semidistributive and necessary and sufficient conditions are specified for its being distributive. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 484–499, July–August, 2006.     

The character table of the group GL 2(Q) when extended by a certain group of order two

LetG denote either of the groupsGL ₂(q) or SL₂(q). Then θ :G →G given by θ(A) = (A ^t)^t, whereA ^t denotes the transpose of the matrixA, is an automorphism ofG. Therefore we may form the groupG.θ> which is the split extension of the groupG by the cyclic group θ of order 2. Our aim in this paper is to find the complex irreducible character table ofG. θ.     

L 1→ L q Poincaré Inequalities for 0 < q < 1 Imply Representation Formulas

Given two doubling measures μ and ν in a metric space (S, ρ) of homogeneous type, let B ₀⊂S be a given ball. It has been a well-known result by now (see [1–4]) that the validity of an L ¹→L ¹ Poincaré inequality of the following form: for all metric balls B⊂B ₀⊂S, implies a variant of representation formula of fractional integral type: for ρ-a.e. x∈B ₀, One of the main results of this paper shows that an L ¹ to L ^q Poincaré inequality for some 0 < q < 1, i.e., for all metric balls B⊂B ₀, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition, also implies the same formula. Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved. Received December 27, 2000, Accepted May 28, 2001     

The character table of a split extension of the Heisenberg group H 1( q ) by Sp (2, q ), q odd

In this paper we determine the full character table of a certain split extension of the Heisenberg group H ₁ by the odd-characteristic symplectic group Sp(2, q).      

Cocycle Twisting of E(n)-Module Algebras and Applications to the Brauer Group

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group BQ(k,E(n)) of E(n) that are isomorphic. For any triangular structure R on E(n) we prove that the subgroup BM(k,E(n),R) of BQ(k,E(n)) arising from R is isomorphic to a direct product of BW(k), the Brauer-Wall group of the ground field k, and Sym_n(k), the group of n × n symmetric matrices under addition. For a general quasi-triangular structure R on E(n) we construct a split short exact sequence having BM(k,E(n),R⁾ as a middle term and as kernel a central extension of the group of symmetric matrices of order r < n (r depending on R). We finally describe how the image of the Hopf automorphism group inside BQ(k,E(n)) acts on Sym_n (k).     

Some group actions on K(x 1 , x 2 , x 3 )

LetK be any field which may not be algebraically closed,K(x ₁ ,x ₂ ,x ₃ ) be the rational function field of three variables overK, and σ:K(x ₁ ,x ₂ ,x ₃ ) → K(x ₁ ,x ₂ ,x ₃ ) be aK-automorphism defined by wherea _i ,b _i ,c _i ,d _i ∈K anda _i d _i −b _i c _i ≠0. Let ,f _i (T)=T ² −(a _i +d _i )T+(a _i d _i −b _i c _i )∈K[T] be the “characteristic polynomial” of σ _i . Theorem:Assume that charK≠2.Then the fixed field K(x ₁ ,x ₂ ,x ₃ ) ^<σ> is not rational (=purely transcendental) over K if and only if (i) for each 1≤i≤3, f _i (T) is irreducible; (ii) the Galois group of f ₁ (T)f ₂ (T)f ₃ (T) over K is of order 8; and (iii) for each 1≤i≤3,ord (σ _[itn] )is an even integer.     

The smooth Banach submanifold B*(E, F) in B(E, F)

Let E,F be two Banach spaces,B(E,F),B+(E,F),Φ(E,F),SΦ(E,F) and R(E,F) be bounded linear,double splitting,Fredholm,semi-Frdholm and finite rank operators from E into F,respectively. Let Σ be any one of the following sets:{T ∈Φ(E,F):Index T=constant and dim N(T)=constant},{T ∈ SΦ(E,F):either dim N(T)=constant< ∞ or codim R(T)=constant< ∞} and {T ∈ R(E,F):Rank T=constant< ∞}. Then it is known that Σ is a smooth submanifold of B(E,F) with the tangent space TAΣ={B ∈ B(E,F):BN(A)-R(A) } for any A ∈Σ. However,for ...     

Switching Sets Regolari E Cubiche Rigate Di P G (5, q ) Piani Di Tralsazione Ad Essi Associati G (5, q ) Piani Di Tralsazione Ad Essi Associati

In a recent paper, the authors studied some algebraic hypersurfaces of the third order in the projective spacePG(5,q) and they called them ruled cubics, since they possess three systems of planes. Any two of these constitute a regular switching set and furthermore, if Σ is a given regular spread ofPG(5,q), one of the three systems is contained in Σ. The subject of this note is to prove, conversely, that every regular switching set (Φ, Φ′) with Φ ⊂ Σ is a ruled cubic and to construct, for a generic choice of the projective reference system inP G(5,q), the quasifield which coordinatizes the translation plane Π associated with the spread (Σ − Φ) ∪ Φ′. The planes Π, of orderq ³, are a generalization of the finite Hall planes.     

A new characterization of L 2(q) by its noncommuting graph

Let G be a non-abelian group and associate a non-commuting 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 this short paper we prove that if G is a finite group with ∇(G) ≅ ∇(M), where M = L ₂(q) (q = p ⁿ , p is a prime), then G ≅ M.