Even degree characters in principal blocks

We characterise finite groups such that for an odd prime p all the irreducible characters in its principal p-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by p unless $p=7$ and the group is $M_{22}$. As a consequence we deduce that if $p\ne 7$ or if $M_{22}$ is not a composition factor of a group G, then the condition above is equivalent to $G/{\mathbf{O}}_{p^{\prime }}(G)$ having odd order.     

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     

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     

Correspondences between constituents of projective characters

Let G be a finite p-solvable group. Let P ∈ Syl _p (G) and N = N _G (P). We prove that there exists a natural bijection between the irreducible constituents of p′-degree of the principal projective character of G and those of . Received: 2 May 2007, Revised: 17 September 2007     

Orthogonality of cosets relative to irreducible characters of finite groups

Studied is an assumption on a group that ensures that no matter how the group is embedded in a symmetric group, the corresponding symmetrized tensor space has an orthogonal basis of standard (decomposable) symmetrized tensors.     

Finite 2-groups in which distinct nonlinear irreducible characters have distinct kernels

In this paper, we study finite 2-groups in which distinct nonlinear irreducible characters have distinct kernels. We prove several results concerning these groups and completely classify 2-groups with at most five nonlinear irreducible characters satisfying this property.     

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.     

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     

On simple modules and normal subgroups of p-solvable groups

Let N be a normal subgroup of a p-solvable group G and let M be a simple FN-module, where F is an algebraically closed field of characteristic p. Next, denote by IRR⁰(FG|M) the set of all simple FG-modules V lying over M such that the p-part of dim_F V is as small as possible. In this paper, |IRR⁰(FG|M)| and the vertices of modules in IRR⁰(FG|M) are determined. The p-blocks of G to which modules in IRR⁰(FG|M) belong are also determined.Received: 5 December 2003     

Clifford classes for isoclinic groups

Let G be a finite group, N a normal subgroup of G, and an irreducible character of G. Clifford Theory studies a whole collection of related irreducible characters of all the subgroups of G that contain N. The relationships among these characters as well as their Schur indices are controlled by the Clifford class c Clif(G/N, F) of with respect to N over some field F. This is an equivalence class of central simple G/N-algebras. Assume now that G/N is cyclic. One can obtain a new isoclinic group and character by multiplying each element of each coset of N in G by an appropriate power of a fixed root of unity . We show that there is a simple formula to calculate the Clifford class of in terms of c and . Hence, the Clifford class c controls not only the Schur index of the characters of all the subgroups of G that contain N, it also controls the Schur indices of the characters of the corresponding characters of the isoclinic groups When is a |G/N|-th root of 1, our formula shows that then When = i and |G/N| = 2, the implicit transformation on Clif(Z/2Z, F) yields a group homomorphism of the group structure introduced on the Brauer-Wall group of F to describe the Schur indices of all the irreducible characters of the double covers of the symmetric and alternating groups.Received: 17 August 2001     

Strongly connective degree sets I: nonabelian solvable quotients

When G is a finite nonabelian group, we associate the common-divisor graph (G) with G by letting nontrivial degrees in cd(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 has a nonabelian solvable quotient, we show that if (G) is connected and has a diameter of at most 2, then indeed cd(G) has a strongly connective subset.Received: 7 July 2004; revised: 5 October 2004     

On a conjecture of huppert for alternating groups

We prove a character degree property of the alternating groups, recently conjectured by Huppert.     

On the category of weak Cayley table morphisms between groups

Weak Cayley table functions between groups are generalized conjugacy-preserving homomorphisms, under which products of images are conjugate to images of products. There is a weak Cayley table bijection between two groups iff they have the same 2-characters. In this paper, weak Cayley table functions are augmented to include the specific conjugating elements, leading to the concept of a weak (Cayley table) morphism. If the conjugating elements are chosen subject to a crossed-product condition, then the weak morphisms between groups form a category. The forgetful functor to this category from the category of group homomorphisms is shown to possess a left adjoint. Two weak morphisms are said to be homotopic if they project to the same weak Cayley table function. As a first step in the analysis of the category of weak morphisms, the group of units of the monoid of weak morphisms homotopic to the identity automorphism of a group is described.     

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     

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     

Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let Un(Fq) denote the group of unipotent n×n upper triangular matrices over a finite field with q elements. We show that the Heisenberg characters of Un+1(Fq) are indexed by lattice paths from the origin to the line x+y=n using the steps (1,0), (1,1), (0,1), (0,2), which are labeled in a certain way by nonzero elements of Fq. In particular, we prove for n?1 that the number of Heisenberg characters of Un+1(Fq) is a polynomial in q−1 with nonnegative integer coefficients and degree n, whose leading coefficient is the nth Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of Un(Fq) is a polynomial in q−1 whose coefficients are Delannoy numbers and whose values give a q-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of Un(Fq) consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in q−1 with nonnegative integer coefficients.     

Basic sets in defining characteristic for general linear groups of small rank

Let q be a power of some prime number p. Let be a connected reductive group defined over the field with q elements and let F be the corresponding Frobenius map. In this note, we give methods to find relations between the restrictions on semisimple elements of the irreducible characters of . As illustration, we explicitly determine a p-basic set for , and .