Frobenius $ \mathbb Q $ -groups

A finite group whose irreducible complex characters are rational is called a -group. In this paper we will find the structure of Frobenius -groups.Received: 14 April 2003     

On distinct character degrees

Berkovich, Chillag and Herzog characterized all finite groups G in which all the nonlinear irreducible characters of G have distinct degrees. In this paper we extend this result showing that a similar characterization holds for all finite solvable groups G that contain a normal subgroup N, such that all the irreducible characters of G that do not contain N in their kernel have distinct degrees.     

The structure of irreducible matrix groups with submultiplicative spectrum

We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p ²× p ² matrices if p is an odd prime.     

Solvable D_2-Groups

Let G be a finite group, Irr_1(G) be the set of nonlinear irreducible characters of G and cd_1(G) the set of degrees of the characters in Irr_1(G). A group G is said to be a D_2-group if |cd_1(G)| =|Irr_1(G)|-2. In this paper, we give a complete classification of solvable D_2-groups.     

A note on character kernels in finite groups of prime power order

In this note, we classify the finite groups of prime power order for which all nonlinear irreducible character kernels constitute a chain with respect to inclusion. Received: 6 April 2007     

Generalisations of conjectures of Brauer and Olsson

We investigate suitable generalisations of the conjectures of Brauer and Olsson bounding numbers of irreducible characters of certain degrees in a block by invariants of the defect group.Received: 5 June 2002     

Groups whose nonlinear irreducible characters separate element orders or conjugacy class sizes

A class function φ on a finite group G is said to be an order separator if, for every x and y in G \ {1}, φ(x) = φ(y) is equivalent to x and y being of the same order. Similarly, φ is said to be a class-size separator if, for every x and y in G\ {1}, φ(x) = φ(y) is equivalent to |C _G (x)| = |C _G (y)|. In this paper, finite groups whose nonlinear irreducible complex characters are all order separators (respectively, class-size separators) are classified. In fact, a more general setting is studied, from which these classifications follow. This analysis has some connections with the study of finite groups such that every two elements lying in distinct conjugacy classes have distinct orders, or, respectively, in which disctinct conjugacy classes have distinct sizes. Received: 10 April 2007     

Groups whose non-linear irreducible characters are rational valued

A finite group G all of whose nonlinear irreducible characters are rational is called a \mathbbQ₁{\mathbb{Q}_1}-group. In this paper, we obtain some results concerning the structure of \mathbbQ₁{\mathbb{Q}_1}-groups.     

Multiplicity-free representations of symmetric groups

Building on work of Saxl, we classify the multiplicity-free permutation characters of all symmetric groups of degree 66 or more. A corollary is a complete list of the irreducible characters of symmetric groups (again of degree 66 or more) which may appear in a multiplicity-free permutation representation. The multiplicity-free characters in a related family of monomial characters are also classified. We end by investigating a consequence of these results for Specht filtrations of permutation modules defined over fields of prime characteristic.     

Above the Glauberman correspondence

Let S be a finite solvable group, and suppose S acts on the finite group N, and they have coprime orders. Then, the celebrated Glauberman correspondence provides a natural bijection from the set IrrS(N) of irreducible characters of N which are invariant under the action of S to the set Irr(CN(S)) of all irreducible characters of the centralizer of S in N. Suppose, further, that the semidirect product SN is a normal subgroup of a finite group G. Let θ∈IrrS(N), and let ψ∈Irr(CN(S)) be its Glauberman correspondent. We prove that there is a bijection with good compatibility properties from the set Irr(G,θ) of the irreducible characters of G above θ to Irr(NG(S),ψ) such that, in the case when S is a p-group for some prime p, it preserves fields of values and Schur indices over Qp, the field of p-adic numbers. Using this result, we also prove a strengthening of the McKay Conjecture for all p-solvable groups.     

Positions of characters in finite groups and the Taketa inequality

We define the position of an irreducible complex character of a finite group as an alternative to the degree. We then use this to define three classes of groups: position reducible (PR)-groups, inductively position reducible (IPR)-groups and weak IPR-groups. We show that IPR-groups and weak IPR-groups are solvable and satisfy the Taketa inequality (ie, that the derived length of the group is at most the number of degrees of irreducible complex characters of the group), and we show that any M-group is a weak IPR-group. We also show that even though PR-groups need not be solvable, they cannot be perfect.     

String C-groups with real Schur index 2

We give examples of finite string C-groups (the automorphism groups of abstract regular polytopes) that have irreducible characters of real Schur index 2. This answers a problem of Monson concerning these groups.     

Finite p-groups all of whose non-abelian proper subgroups are metacyclic

In this paper, we classify the finite p-groups all of whose non-abelian proper subgroups are metacyclic and answer a question posed by Berkovich. Received: 22 June 2005     

有限群的特征标的零点和可解φ-群

有两个对偶的问题如下：问题Ⅰ：将满足下述条件的有限群G分类：G的特征标表中，除一行外其余各行最多有一个零．问题Ⅱ：将满足下述条件的有限群G分类：G的特征标表中，除一列外其余各列最多有一个零．在这篇文章中，我们对于有限可解群解答上述两个问题，并确定和这两个问题密切相关的一类有限可解群的结构（这类可解群在本文中称之为可解φ-群）．附带我们还完全回答了[4]中的问题1，并说明[6，定理]的条件可以极大地减弱．     

Constituents of supercharacters and Kirillov functions

Supercharacters and Kirillov functions have been proposed as two related constructions that mimic the characters of certain p-groups. The irreducible constituents of these functions are intimately linked; specifically, irreducible constituents of a Kirillov function are constituents of the corresponding supercharacter.     

Arithmetic of characters of generalized symmetric groups

The result here answers the following questions in the affirmative: Can the Galois action on all abelian (Galois) fields $K/\mathbb{Q}$ be realized explicitly via an action on characters of some finite group? Are all subfields of a cyclotomic field of the form $\mathbb{Q}(\chi)$, for some irreducible character $\chi$ of a finite group G? In particular, we explicitly determine the Galois action on all irreducible characters of the generalized symmetric groups. We also determine the smallest extension of $\mathbb{Q}$ required to realize (using matrices) a given irreducible representation of a generalized symmetric group. Received: 18 February 2002     

Vertices of simple modules and normal subgroups ofp-solvable groups

Let π be a set of prime numbers andG a finite π-separable group. Let θ be an irreducible π′-partial character of a normal subgroupN ofG and denote by I_π′ (G‖θ), the set of all irreducible π′-partial characters Φ ofG such that θ is a constituent of Φ_N. In this paper, we obtain some information about the vertices of the elements in I_π′ (G‖θ). As a consequence, we establish an analogue of Fong's theorem on defect groups of covering blocks, for the vertices of the simple modules (in characteristicsp) of a finitep-solvable group lying over a fixed simple module of a normal subgroup.     

Finite 2-groups all of whose nonmetacyclic subgroups are generated by involutions

We classify here nonmetacyclic finite 2-groups all of whose nonmetacyclic subgroups are generated by involutions (Theorem 1.1). This solves problem Nr. 710 for p = 2 stated by Y. Berkovich in [1]. For p > 2 the corresponding problem is open. Received: 14 March 2007 Revised: 15 April 2007     

Brauer characters with cyclotomic field of values

It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675-686] that, for any odd prime p, every finite group of even order has a non-trivial rational-valued irreducible p-Brauer character. For p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠q are primes, then any finite group of order divisible by q has a non-trivial irreducible p-Brauer character with values in the cyclotomic field Q(exp(2πi/q)).