A construction of two distinct canonical sets of lifts of Brauer characters of a p-solvable group

In [5], Navarro defines the set , where Q is a p-subgroup of a p-solvable group G, and shows that if δ is the trivial character of Q, then Irr(G|Q, δ) provides a set of canonical lifts of IBr_p(G), the irreducible Brauer characters with vertex Q. Previously, in [2], Isaacs defined a canonical set of lifts B_π(G) of I_π(G). Both of these results extend the Fong-Swan Theorem to π-separable groups, and both construct canonical sets of lifts of the generalized Brauer characters. It is known that in the case that 2∈π, or if |G| is odd, we have B_π(G) = Irr(G|Q, 1_Q). In this note we give a counterexample to show that this is not the case when . It is known that if and χ∈B_π(G), then the constituents of χ_N are in B_π (N). However, we use the same counterexample to show that if , and χ∈Irr(G|Q, 1_Q) is such that θ ∈Irr(N) and [θ, χ _N] ≠ 0, then it is not necessarily the case that θ ∈Irr(N) inherits this property. Received: 17 October 2005     

The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity

Maximum set of edges no two covered by a clique

Leth(G) be the largest number of edges of the graphG. no two of which are contained in the same clique. ForG without isolated vertices it is proved that ifh(G)≦5, thenχ( )≦h(G), but ifh(G)=6 thenχ( ) can be arbitrarily large.     

Fong characters and chains of normal subgroups

In this note, we show that if is a π-partial character of the π-separable group is a chain of normal subgroups of G, and H is a Hall π-subgroup of G, then has a Fong character α Irr(H) such that for every subgroup , every irreducible constituent of α _H∩N is Fong for N. We also show that if is quasi-primitive, then for every normal subgroup M of G the irreducible constituents of are Fong for M. Received: 21 July 2006 Revised: 17 January 2007     

The minimum degree threshold for perfect graph packings

Let H be any graph. We determine up to an additive constant the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let δ(H,n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G)≥k contains a perfect H-packing. We show that

The Littlewood-Gowers problem

The paper has two main parts. To begin with, suppose that G is a compact abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions ƒ ∈ L ²(G). We prove an analogous result for functions ƒ ∈ A(G), where A(G) is the space endowed with the norm , and generalize this to the approximate Fourier transform on Bohr sets. As an application of the first part of the paper, we improve a recent result of Green and Konyagin. Suppose that p is a prime number and A ⊂ ℤ/pℤ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/3−ɛ; we improve this to ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/2−ɛ. To put this in context, it is easy to see that if A is an arithmetic progression, then ‖χ _A ‖ _A(ℤ/pℤ) ≪ log p.     

On a subalgebra of the centre of a group ring II

Let p be a prime, G a finite group with p | |G| and F a field of characteristic p. By we denote the F-subspace of the centre of the group ring FG spanned by the p-regular conjugacy class sums. J. Murray proved that is an algebra, if G is a symmetric or alternating group. This can be used for the computation of the block idempotents of FG. We proved that is an algebra if the Sylow-p-subgroups of G are abelian. Recently, Y. Fan and B. Külshammer generalized this result to blocks with abelian defect groups. Here, we show that is an algebra if the Sylow-2-subgroups of G are dihedral. Therefore and are algebras for all primes p and all prime powers q. Furthermore we prove that is an algebra for the simple Suzuki-groups Sz(q), where q is a certain power of 2 and p is an arbitrary prime dividing |Sz(q)|. Received: 18 May 2007     

Gauss Sum of Index 4： （2） Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.     

On central automorphisms that fix the centre elementwise

Let G be a group and let Aut _c (G) be the group of central automorphisms of G. Let be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper we prove that if G is a finite p-group, then = Inn(G) if and only if G is abelian or G is nilpotent of class 2 and Z(G) is cyclic. This work was supported in part by the Center of Excellence for Mathematics, University of Isfahan, Iran. Received: 30 October 2006     

Exact asymptotic behaviour of the codimensions of some P.I. algebras

Letc _n (A) denote the codimensions of a P.I. algebraA, and assumec _n (A) has a polynomial growth: . Then, necessarily,q∈ℚ [D3]. If 1∈A, we show that , wheree=2.71…. In the non-unitary case, for any 0<q∈ℚ, we constructA, with a suitablek, such that . In memory of S. A. Amitsur, our teacher and friend Partially supported by Grant MM404/94 of Ministry of Education and Science, Bulgaria and by a Bulgarian-American Grant of NSF. Partially supported by NSF grant DMS-9101488.     

Strongly connective degree sets II: perfect derived subgroups

When G is a finite nonabelian group, we associate the common-divisor graph with G by letting nontrivial degrees in cd(G) = {χ(1) | χ∈Irr(G)} be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of , and every vertex outside of is adjacent to some member of . When G is nonsolvable, we provide sufficiency conditions for cd(G) to have a strongly connective subset. We also extend a previously known result about groups with nonabelian solvable quotients, and prove for arbitrary groups G that if the associated graph is connected and has a diameter bounded by 2, then indeed cd(G) has a strongly connective subset. The major focus is on when the derived subgroup G′ is perfect. Received: 23 July 2005     

Kummer's lemma for cyclic 2-groups

For cyclic 2-groupsC, we characterize the kernel of the map induced on the units of the integral group ring by the coefficient reduction . This allows us to prove that, for any finite abelian 2-groupA, the circular units of ℤA (i.e. those which are mapped to cyclotomic units by every character ofA) can be generated in a certain systematic way. Work supported in part by an NSERC (Canada) operating grant     

The computations of some Schur indices

Let χ be an irreducible character of a finite groupG. Letp=∞ or a prime. Letm _p (χ) denote the Schur index of χ overQ _p , the completion ofQ atp. It is shown that ifx is ap′-element ofG such that for all irreducible charactersX _u ofG thenm _p (χ)/vbχ(x). This result provides an effective tool in computing Schur indices of characters ofG from a knowledge of the character table ofG. For instance, one can read off Benard’s Theorem which states that every irreducible character of the Weyl groupsW(E _n), n=6,7,8 is afforded by a rational representation. Several other applications are given including a complete list of all local Schur indices of all irreducible characters of all sporadic simple groups and their covering groups (there is still an open question concerning one character of the double cover of Suz). This work was partly supported by NSF Grant MCS-8201333.     

Lifting Unitary Elements with Application to Approximately Inner Automorphisms

Let A be a separable unital nuclear simple C＊-algebra with torsion K0 （A）, free K1 （A） and with the UCT. Let T ： A→M（K）/K be a unital homomorphism. We prove that every unitary element in the commutant of T（A） is an exponent, thus it is liftable. We also prove that each automorphism α on E with α ∈ Aut0（A） is approximately inner, where E is a unital essential extension of A by K and α is the automorphism on A induced by α.     

Fields of values of Fong characters

If G is a p-solvable finite group with a p-complement H and φ ∈ IBr(G), then P. Fong showed that there exists α ∈ Irr(H) such that α^G=Φ_φ. In this note we prove that α can be chosen such that the field of values index divides φ (1)p. Received: 6 May 2005     

Some results on ℚ-groups

A finite group G whose irreducible characters are rational valued is called a ℚ-group. In this paper we will be concerned with the structure of a finite ℚ-group that contains a strongly embedded subgroup and the structure of a finite ℚ-group satisfying the property that none of its sections is isomorphic to .      

On automorphisms of A-groups

Let G be an A-group (i.e. a group in which xx ^α  = x ^α x for all and let denote the subgroup of Aut(G) consisting of all automorphisms that leave invariant the centralizer of each element of G. The quotient is an elementary abelian 2-group and natural analogies exist to suggest that it might always be trivial. It is shown that, in fact, for any odd prime p and any positive integer r, there exist infinitely many finite pA-groups G for which has rank r. Received: 23 March 2008, Revised: 20 May 2008     

Lifts, Discrepancy and Nearly Optimal Spectral Gap*

We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map π :H →G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n “new” eigenvalues. We conjecture that every d-regular graph has a 2-lift such that all new eigenvalues are in the range (if true, this is tight, e.g. by the Alon–Boppana bound). Here we show that every graph of maximal degree d has a 2-lift such that all “new” eigenvalues are in the range for some constant c. This leads to a deterministic polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue . The proof uses the following lemma (Lemma 3.3): Let A be a real symmetric matrix with zeros on the diagonal. Let d be such that the l₁ norm of each row in A is at most d. Suppose that for every x,y ∈{0,1}ⁿ with ‹x,y›=0. Then the spectral radius of A is O(α(log(d/α)+1)). An interesting consequence of this lemma is a converse to the Expander Mixing Lemma. * This research is supported by the Israeli Ministry of Science and the Israel Science Foundation.     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

Group Algebras: Normal Subgroups and Ideals

The standard correspondence between the normal subgroups of the group G and some ideals of the group algebra FG is described. There is the problem of what we can say (or even prove) about a two-sided ideal of that does not contain any element of the form 1 − g ≠ 0, g ∈G of the standard basis of the augmentation ideal of . The main part of the argument of [2] yields the insight that, for such an ideal I there exists an expansion such that the ideal J of spanned by I contains an element 1 − h, h ∈ H \ G. Using the ideas of [2], we construct -thick groups H such that for every ideal J ≠ (0) of there are elements 1 − h ≠ 0 in J. This construction allows many variations. Examples of simple -thick groups were pointed out in [2]. A natural class of (in general non-simple) -full groups are the normal sections of the groups