Two Remarks on the Glauberman-Isaacs Character Correspondence

Let the finite group A act as an automorphism group on the finite group G. When (¦G¦,¦A¦) = 1,we have the Glauberman-Isaacs natural character correspondence (bijection) *: Irr(G)^A (the A-fixed irreducible characters of (G)) → Irr(C_G(A)) (the irreducible characters of C_G(A)). We present a short proof of a Theorem of G. Navarro ([9, Theorem A]), and a reduction of the general conjecture that Χ*(1) divides Χ(1) for all Χ ∈ Irr(G)^A to the verification of this conjecture in which G is quasi-simple.     

Finite groups and subgroup-permutability

Subgrouppermutability in a finite group is studied, in order to distinguish between irreducible and reducible groups and to give a bound for the diameter. A simple nonabelian finite group is irreducible and its diameter is less or equal16 while a reducible group is forced to be soluble and the bound of the diameter of such a group is 4.Research partially supported by G.N.S.A.G.A. of C.N.R. and M.U.R.S.T. of Italy.     

关于Glauberman-Isaacs特征标对映的注记

假设群A经自同构互素地作用在G上.设χ是G的一个A-不变不可约特征标,π(G,A)表示Glauberman-Isaacs特征标对映.对于B≤A,T.R.Wolf曾猜想χπ(G,A)是χπ(G,B)a的一个不可约成份,此处C=CG(A).设G=N(X)H且(|N|,|H|)=1,假定H是A-不变的且N是一个Sylow塔群,N的Sylow-子群是交换的.在本文中,我们证明了如果这个猜想对所有H的A-不变子群成立,则猜想对G也成立.     

A Note on the Subdirectly Irreducible ckham Algebras

ＡＮｏｔｅｏｎｔｈｅＳｕｂｄｉｒｅｃｔｌｙＩｒｒｅｄｕｃｉｂｌｅｃｋｈａｍＡｌｇｅｂｒａｓ￥ＦａｎｇＪｉｅ（Ｄｅｐａｒｔｍｅｎｔｏｆｍａｔｈｅｍａｔｉｃｓ，ＺｈｏｎｇｓｈａｎＵｎｉｖｅｒｓｉｔｙ，Ｇｕａｎｇｚｈｏｕ，５１０２７５）Ａｂｓｔｒａｃｔ：Ｔ...     

Extension of Hardy–Littlewood–Sobolev Inequalities for Riesz Potentials on Hypergroups

Mediterranean Journal of Mathematics - Let p be a prime number, let G be a finite group, let N be a normal subgroup of G, and let $$\theta $$ be a G-invariant irreducible character of N. In Rizo (J...     

Taketa’s theorem for some character degree sets

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then ${{\rm dl}(G) \leqslant |{\rm cd} (G)|}$ , where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality ${{\rm dl}(N) \leqslant |{\rm cd} {(G \mid N)}|}$ holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if ${{\rm cd} {(G \mid N')}}$ is a set of non-trivial p–powers for some fixed prime p.     

Extensions and the Induced Characters of Cyclic p-Groups

For a prime p, we denote by B_n the cyclic group of order pⁿ. Let φ be a faithful irreducible character of B_n, where p is an odd prime. We study the p-group G containing B_n such that the induced character φ^G is also irreducible. The purpose of this article is to determine the subgroup N_G(N_G(B_n)) of G under the hypothesis [N_G(B_n):B_n]⁴ ≦ pⁿ.     

An upper bound for the order of primitive permutation groups

Let G be a primitive permutation group of order |G| and degree n. Then |G|≤nd^m, where d is the minimal size of a nontrivial orbit of a one-point stabilizer of G and m is the minimal degree of a nonprincipal irreducible representation of G entering its permutation representation. Bibliography: 8 titles. Dedicated to L. D. Faddeev on occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 256–263. Translated by I. Ponomarenko.     

Finite groups with Frobenius subgroup

Suppose the normalizer N of a subgroup A of a simple group G is a Frobenius group with kernel A, and the intersection of A with any other conjugate subgroup of G is trivial, and suppose, if A is elementary Abelian, that ¦a¦> 2n+1, where n=¦N:A¦. It is proved that if A has a complement B in G, then G acts doubly transitively on the set of right cosets of G modulo B, the subgroup B is maximal in G, and ¦B¦ is divisible by ¦a¦–1. The proof makes essential use of the coherence of a certain set of irreducible characters of N.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 177–186, August, 1976.The author would like to thank V. D. Mazurov for helpful discussions concerning the theorem proved in this note.     

THE ACTION OF THE LINEAR CHARACTER GROUP ON THE SET OF NON-LINEAR IRREDUCIBLE CHARACTERS THE ACTION OF THE LINEAR CHARACTER GROUP ON THE SET OF NON-LINEAR IRREDUCIBLE CHARACTERS

1. Notations and Basic ResultsLet G be a finite nonabelian group. Then frs(G/G') is an abelian group under themultiplication of characters and acts on the set of non-linear irreducible characters of G viathe multiplication of characters. The purpose of this paper is to investigate this action. Asan application of our theoryl in the end of Section 3 we give the classification of groupshaving exactly three non-linear irreducible caracters.All groups in the paper are finite. For a factor grou…     

On a composition of subfactors with group subfactors

Let M be a type Ⅱ_1 factor,G be a finite group,and N ■ M be an irreducible subfactor of finite index.We prove that the composed lattice of the intermediate subfactor lattice for the inclusion N ■ M and the subgroup lattice of G can be realized as an intermediate subfactor lattice of a certain composed subfactor of finite index,and this subfactor also has finite depth when N ■ M has finite depth.     

The Computation of Invariant Fields and a Constructive Version of a Theorem by Rosenlicht

Let G be an algebraic group acting on an irreducible variety X. We present an algorithm for computing the invariant field k(X)^G. Moreover, we give a constructive version of a theorem of Rosenlicht, which says that almost all orbits can be separated by rational invariants. More precisely, we give an algorithm for computing a nonempty open subset of X with a geometric quotient.     

Transitive groups with irreducible representations of bounded degree

A well-known theorem of Jordan states that there exists a function J(d) of a positive integer d for which the following holds: if G is a finite group having a faithful linear representation over ℂ of degree d, then G has a normal Abelian subgroup A with [G:A]≤J(d). We show that if G is a transitive permutation group and d is the maximal degree of irreducible representations of G entering its permutation representation, then there exists a normal solvable subgroup A of G such that [G:A]≤J(d) ^log ₂ ^d. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 108–119. Translated by S. A. Evdokimov.     

特征标次数的重数与可解群结构

非线性不可约特征标次数的重数全部为1的有限群的分类是熟知的.对可解群,本文讨论更一般的,即非线性不可约特征标次数的重数都与群阶互素的有限群的纯群论性质.特别地,得到了非线性不可约特征标次数的重数均小于2p的奇阶群G的分类结果.这里p为群阶|G|的最小素因子.     

On the control of fusion in the local category for the p-block with a minimal nonabelian defect group

If B is a p-block of a finite group G with a minimal nonabelian defect group D (p is an odd prime number) and (D, b D ) is a Sylow B-subpair of G, then N G (D, b D ) controls B-fusion of G in most cases. This result is of great importance, because we can use it to obtain a complete set of representatives of G-conjugate classes of B-subsections and to calculate the number of ordinary irreducible characters in B. This result is key to the calculation of the structure invariants of the block with a minimal non...     

The supersingular locus of the Shimura variety of GU(1,<Emphasis Type="Italic">n</Emphasis>−1) II

We complete the study of the supersingular locus M^ss\mathcal{M}^{\mathrm{ss}} in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ in the case that p is inert. This was started by the first author in Can. J. Math. 62, 668–720 (2010) where complete results were obtained for n=2,3. The supersingular locus M^ss\mathcal{M}^{\mathrm{ss}} is uniformized by a formal scheme N\mathcal{N} which is a moduli space of so-called unitary p-divisible groups. It depends on the choice of a unitary isocrystal N. We define a stratification of N\mathcal{N} indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of N. We show that the combinatorial behavior of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the irreducible components of N_red\mathcal{N}_{\mathrm{red}}) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that M^ss\mathcal{M}^{\mathrm{ss}} is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of M^ss\mathcal{M}^{\mathrm{ss}}.     

关于Brauer特征标扩张的一点注记

研究有限群特征标可扩张的情况是有限群表示论领域中一个有意义的问题.设G为有限群,用Irr（G）表示G的所有不可约复特征标构成的集合.设N（？）G,θ∈Irr（N）且θ是G-不变.如果（｜G/N｜,o（θ）θ（1））=1,则[1]中的推论8.16说明了此时υ到G有唯一的扩张χ,且o（χ）=o（θ）.此结论启发了我们可以从特征标的行列式阶的角度来思考特征标扩张的情形.本文将利用有限群Brauer特征标的行列式阶,着重考虑Brauer特征标的可扩张情形.另外我们也得到了一个关于Brauer特征标次数的结论.     

Globally Irreducible Representations of Finite Groups and Integral Lattices

The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L ₂(q) and ²B₂(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pⁿ-1)/2 of the symplectic group Sp_2n(p) (p 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pⁿ–1). A summary of currently known globally irreducible representations is given.     

An algorithm and self-reproducing systems

Let G be a group and let N be a normal subgroup of G. We set cd(G|N) to be the degrees of the irreducible characters of G whose kernels do not contain N. We associate a graph with this set. The vertices of this graph are the primes dividing degrees in cd(G|N), and there is an edge between p and q if pq divides some degree in cd(G|N). In this paper, we study this graph when it is disconnected, and we study its diameter when it is connected.