Finite groups in whichp′-classes haveq′-length

IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO ^q′ (G) is solvable,l _q (G)≤1 andl _p (O ^q′ (G)) ≤2. Further, the structure ofG is determined to some extent. Work partially supported by MURST research program “Teoria dei gruppi ed applicazioni”.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Derived subgroups of fixed points

LetA be an elementary abelianq-group acting on a finiteq′-groupG. We show that ifA has rank at least 3, then properties ofC _G(a)′, 1 ≠a ∈A restrict the structure ofG′. In particular, we consider exponent, order, rank and number of generators. This author was supported by the NSF. This author was supported by CNP_q-Brazil.     

Block Primitive 2- (v, k, 1) Designs Admitting a Ree Simple Group

In this paper, we prove that if D is a 2- (v, k, 1) design withG  ≤  Aut(D) block primitive and soc (G)  =  ²G₂(q) then D is a Ree unital with parameters 2- (q³ +  1, q +  1, 1).     

Equivalence classes of matrices inGL 2 (Q) andSL 2 (Q)

LetG denote either of the groupsGL ₂ (q) orSL ₂ (q). The mapping θ sending a matrix to its transpose-inverse is an automophism ofG and therefore we can form the groupG ⁺ =G. <θ>. In this paper conjugacy classes of elements inG ⁺ -G are found. These classes are closely related to the congruence classes of invertible matrices inG.     

Basic sets of Brauer characters of finite groups of Lie type,III

LetG(F ^q ) be a finite classical group whereq is odd and the centre ofG is connected. We show that there exists a set of irreducible characters ofG(F ^q ) such that the corresponding matrix of scalar products with the characters of Kawanaka’s generalized Gelfand-Graev representations is square unitriangular. This uses in an essential way Lusztig’s theory of character sheaves. As an application we prove that there exists an ordinary basic set of 2-modular Brauer characters and that the decomposition matrix of the principal 2-block ofG(F ^q ) has a lower unitriangular shape.     

Schur functions and the invariant polynomials characterizing U(n) tensor operators

We give a direct formulation of the invariant polynomials _μG_q⁽ⁿ⁾(, Δ_i,;, x_{i,i + 1},) characterizing U(n) tensor operators p, q, …, q, 0, …, 0 in terms of the symmetric functions S_λ known as Schur functions. To this end, we show after the change of variables Δ_i = γ_i − δ_i and x_{i, i + 1} = δ_i − δ_{i + 1} that_μG_q⁽ⁿ⁾(,Δ_i;, x_{i, i + 1},) becomes an integral linear combination of products of Schur functions S_α(, γ_i,) · S_β(, δ_i,) in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}, respectively. That is, we give a direct proof that _μG_q⁽ⁿ⁾(,Δ_i,;, x_{i, i + 1},) is a bisymmetric polynomial with integer coefficients in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials _μ^mG_q⁽ⁿ⁾(γ₁,…, γ_n; δ₁,…, δ_m). These new symmetries enable us to give an explicit formula for both _μ^mG₁⁽ⁿ⁾(γ; δ) and ₁G₂⁽ⁿ⁾(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for _μ^mG_q⁽ⁿ⁾(γ; δ).     

Reduction theory for a rational function field

LetG be a split reductive group over a finite field F_q. LetF = F_q(t) and let A denote the adèles ofF. We show that every double coset inG(F)/G(A)/K has a representative in a maximal split torus ofG. HereK is the set of integral adèlic points ofG. WhenG ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.     

On a class of transitive permutation groups of prime degreep=4n+1

SupposeG is a nonsolvable transitive permutation group of prime degreep, such that |N _G v(P)|=p(p−1) for some Sylowp-subgroupP ofG. Letq be a generator of the subgroup ofN _G (P), fixing one letter (it is easy to show that this subgroup is cyclic). Assume thatG contains an elementj such thatj ⁻¹ qj=q ^(p+1)/2. We shall prove that for almost all primesp of the formp=4n+1, a group that satisfies the above conditions must be the symmetric group on a set withp elements.     

Direct factors of polynomial rings over finite fields

Let G_q denote the multiplicative semigroup of all monic polynomials in one indeterminate over a finite field F_q with q elements. By a direct factor of G_q is understood a subset B₁ of G_q such that, for some subset B₂ of G_q, every polynomial w G_q has a unique factorization in the form w = b₁b₂ for b_i B_i. An asymptotic formula B₁^#(n) c₁qⁿ as n → ∞ is derived for the total number B₁^#(n) of polynomials of degree n in an arbitrary direct factor B₁ of G_q, c₁ a constant depending on B₁.     

Spanning trees and a conjecture of Kontsevich

Kontsevich conjectured that the number of zeros over the fieldF _q of a certain polynomialQ _G associated with the spanning trees of a graphG is a polynomial function ofq. We show the connection between this conjecture, the Matrix-Tree Theorem, and orthogonal geometry. We verify the conjecture in certain cases, such as the complete graph, and discuss some modifications and extensions.Partially supported by NSF grant #DMS-9743966.     

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

LetC=C(C, P, k) be the coordinate ring of the affine curve obtained by removing a closed pointP from a (suitable) projective curveC over afinite fieldk. Let SL₂ (C,q) be the principal congruence subgroup of SL₂(C) andU ₂(C,q) be the subgroup generated by the all unipotent matrices in SL₂(C,q), whereq is aC-ideal. In this paper we prove that, for all but finitely manyq, the quotient SL₂(C,q)/U ₂(C,q) is a free group of finite,unbounded rank. LetC(SL₂(A)) be the congruence kernel of SL₂(A), whereA is an arithmetic Dedekind domain with only finitely many units. (e.g.A=C or ℤ) and letG be any finitely generated group. From the above (and previous results) we deduce that the profinite completion ofG,Ĝ, is a homonorphic image ofC(SL₂(A)). This is related to previous results of Lubotzky and Mel'nikov.     

A characterization of PGL(2,q),q ODD

LetG be a finite group andA andB solvable subgroups ofG, such thatG=AB and 2 is the only common prime divisor ofA andB. Under suitable restrictions of the 2-structure ofA andB, it is shown that eitherG is solvable orG contains a solvable normal subgroupN, such thatG/N contains a normal subgroup, which is isomorphic to PGL(2,q),q odd.     

Degenerate principal series representations of Sp(<Emphasis Type="Italic">p,q</Emphasis>)

Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL _q (ℍ)×Sp(p−q). Forsεℂ andμ a highest weight of Sp(p−q), let п_s,μ be the representation ofP such that its restriction toN is trivial and ⊠T _p-q ^μ , where det _q is the determinant character of GL _q (ℍ) andT _p-q ^μ is the irreducible representation of Sp(p−q) with highest weightμ. LetI _p,q(s, μ) be the Harish-Chandra module of the induced representation Ind _P ^G . In this paper, we shall determine the module structure and unitarity ofI _{p, q}(s, μ). Partially supported by NUS grant R-146-000-026-112.     

The trivial intersection problem for characters of principal indecomposable modules

For a prime p and a finite group G let Φ_p(G) denote the complex character associated to the projective indecomposable module in characteristic p with trivial head. Let Irr(Φ_p(G)) denote the set of irreducible characters occurring as constituents in Φ_p(G). We characterize all finite simple groups which satisfy Irr(Φ_p(G))∩Irr(Φ_q(G))={1_G} for all primes p≠q.     

G-Loops and Permutation Groups

A G-loop is a loop which is isomorphic to all its loop isotopes. We apply some theorems about permutation groups to get information about G-loops. In particular, we study G-loops of order pq, where p < q are primes and p  (q − 1). In the case p = 3, the only G-loop of order 3q is the group of order 3q. The notion “G-loop” splits naturally into “left G-loop” plus “right G-loop.” There exist non-group right G-loops and left G-loops of order n iff n is composite and n > 5.     

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.     

Groups with strongly self-centralizing 3-centralizers

LetG be a finite group with a subgroupM which is the centralizer inG of each of its nonidentity elements and let 3 divide the order ofM. Such groups are classified under the assumption that eitherq=[N _G (M):M]=2^s;s≧0 ands ≠ 3 orq<8. This paper was written while the author was visiting the Department of Mathematics, University of Illinois, Urbana, Illinois 61801.     

P.I.G-rings and the contractability of primes

LetR=F{x ₁, …, x_k} be a prime affine p.i. ring andS a multiplicative closed set in the center ofR, Z(R). The structure ofG-rings of the formR _s is completely determined. In particular it is proved thatZ(R _s)—the normalization ofZ(R _s) —is a prüfer ring, 1≦k.d(R _s)≦p.i.d(R _s) and the inequalities can be strict. We also obtain a related result concerning the contractability ofq, a prime ideal ofZ(R) fromR. More precisely, letQ be a prime ideal ofR maximal to satisfyQϒZ(R)=q. Then k.dZ(R)/q=k.dR/Q, h(q)=h(Q) andh(q)+k.dZ(R)/q=k.dz(R). The last condition is a necessary butnot sufficient condition for contractability ofq fromR.     

Galois groups ofp-extensions with two ramification points

LetK _p (p, q) be the maximalp-extension of the field ℚ of rational numbers with ramification pointsp andq. LetG _p (p, q) be the Galois group of the extensionK _p(p.q)/ℚ. It is known thatG _p(p, q) can be presented by two generators which satisfy a single relation. The form of this relation is known only modulo the second member of the descending central series ofG _p(p, q). In this paper, we find an arithmetical-type condition on which the form of the relation modulo the third member of the descending central series ofG _p(p, q) depends. We also consider two examples withp=3,q=19 andp=3,q=37. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 48–60, January–March, 2000. Translated by H. Markšaitis