Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $χ\in \rm{Irr}(G)$, the number $\rm{cod}(χ):=|G:\rm{ker}χ|/χ(1)$ is called the co-degree of $χ$. The set of co-degrees of all irreducible characters of $G$ is denoted by $\rm{cod}(G)$. In this paper, we show that for a non-trivial finite group $G$, $|\rm{Irr}(G)|=|\rm{cod}(G)|$ if and only if $G$ is isomorphic to the cyclic group $\mathbb{Z}_2$ or the symmetric group $S_3$.
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.
Let G be a finite p-group, for some prime p, and $ψ, θ\in \Irr(G)$ be irreducible complex characters of G. It has been proved that if, in addition, $ψ,θ$ are faithful characters, then the product $ψθ$ is a multiple of an irreducible or it is the nontrivial linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k>0, we can always find a p-group G and irreducible characters $Ψ$ and $Θ$ such that $ΨΘ$ is the nontrivial combination of exactly k distinct irreducible characters. We d…
A character of a group is said to be super-monomial if every primitive character inducing it is linear. It is conjectured by Isaacs that every irreducible character of an odd $M$-group is super-monomial. We show that all non linear irreducible characters of lowest degree of an odd $M$-group is super-monomial and provide cases in which one can guarantee that certain irreducible characters of normal subgroups are super-monomial. Finally, we study groups having two irreducible monomial character degrees.
We classify the finite groups whose non-linear irreducible characters that are not conjugate under the natural Galois action have distinct degrees, therefore extending the results in Berkovich et al. [Proc. Amer. Math. Soc. {\bf 115} (1992), 955-959] and Dolfi et al. [Israel J. Math. {\bf 198} (2013), 283-331].
We study the solvable groups $G$ that have an irreducible character $χ\in \Irr(G)$ such that $χ\barχ$ has at most two non-principal irreducible constituents.
We classify the finite groups $G$ which satisfies the condition that every complex irreducible character,whose degree's square doesn't divide the index of its kernel in $G$, lies in the same Galois conjugacy class.
In this work, we classify all finite groups such that for every field extension F of \mathbb{Q}, F is the field of values of at most 3 irreducible characters.
In this paper we determine the ordinary irreducible characters of the five-dimensional full linear group over a Galois field of q elements. We use the techniques developed by J. A. Green.
Let G be a finite group and N be a non-trivial normal subgroup of G, such that the average character degree of irreducible characters in Irr(G|N) is less than or equal to 16=5. Then we prove that N is solvable. Also, we prove the solvability of G, by assuming that the average character degree of irreducible characters in Irr(G|N) is strictly less than 16=5. We show that the bounds are sharp.
For an irreducible character $χ$ of a finite group $G$, the codegree of $χ$ is defined as $|G:\ker(χ)|/χ(1)$. In this paper, we determine finite nonsolvable groups with exactly three nonlinear irreducible character codegrees, and they are $\mathrm{L}_2(2^f)$ for $f\ge 2$, $\mathrm{PGL}_2(q)$ for odd $q\ge 5$ or $\mathrm{M}_{10}$.
Let $G$ be a finite nilpotent group, $χ$ and $ψ$ be irreducible complex characters of $G$ of prime degree. Assume that $χ(1)=p$. Then either the product $χψ$ is a multiple of an irreducible character or $χψ$ is the linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters.
We consider groups where the centers of the irreducible characters form a chain. We obtain two alternate characterizations of these groups, and we obtain some information regarding the structure of these groups. Using our results, we are able to classify those groups where the kernels of the irreducible characters form a chain. We show that a result of Nenciu regarding nested GVZ groups is really a result about nested groups. We obtain an alternate proof of a theorem of Isaacs regarding the existence of $p$-groups with a given set of irreducible character degrees.
Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the Alperin--McKay conjecture.
Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if $G$ is an $M$-group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if $G$ is an $M$-group of odd order where every irreducible character is a $\{p \}$-lift for some prime $p$. We say that a group where irreducible character is super monomial is a super $M$-group. We use our results to find an example of a super $M$-group that has a subgroup that is not a super $M$-group.
Let $G$ be a finite $p$-group and $χ,ψ$ be irreducible characters of $G$. We study the character $χψ$ when $χψ$ has at most $p-1$ distinct irreducible constituents.
In this paper we consider finite groups G satisfying the following condition: G has two columns in its character table which differ by exactly one entry. It turns out that such groups exist and they are exactly the finite groups with a non-trivial intersection of the kernels of all but one irreducible characters or, equivalently, finite groups with an irreducible character vanishing on all but two conjugacy classes. We investigate such groups and in particular we characterize their subclass, which properly contains all finite groups with non-linear characters of distinct degrees, which were ch…
Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on several earlier related results.
In this paper, we prove the existence of a relation between the prime divisors of the order of a Sylow normalizer and the degree of characters having values in some small cyclotomic fields. This relation is stronger when the group is solvable.
In this paper, we construct a group with three real irreducible characters whose Sylow 2-subgroup is an iterated central extension of a Suzuki 2-group. This answers a question raised by Moreto and Navarro who asked whether such a group exists.
The branching theorem expresses irreducible character values for the symmetric group $S_n$ in terms of those for $S_{n-1}$, but it gives the values only at elements of $S_n$ having a fixed point. We extend the theorem by providing a recursion formula that handles the remaining cases. It expresses these character values in terms of values for $S_{n-1}$ together with values for $S_n$ that are already known in the recursive process. This provides an alternative to the Murnaghan-Nakayama formula.
We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. We find a bound that depends on the prime $p$. If we further restrict our average by fixing a subfield $k$ of the complex numbers and then compute the average of the degrees of the irreducible characters whose degrees are not divisible by $p$ and have values in $k$, then we will see that we obtain a bound that depends on both $p$ and $k$. Moreover, we find examples that make tho…
Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $χ(g)\neq 0$ for all irreducible characters $χ$ of $G$. Such an element is said to be non-vanishing in $G$. Let $p$ be a prime. If all $p$-elements of $N$ satisfy the previous property, then we prove that $N$ has a normal Sylow $p$-subgroup. As a consequence, we also study certain arithmetical properties of the $G$-conjugacy class sizes of the elements of $N$ which are zeros of some irreducible character of $G$. In particular, if $N=G$, then new contributions are obtained.
In this paper, we determine new characterizations of nested and nested GVZ-groups, including character-free characterizations, but we additionally show that nested groups and nested GVZ-groups can be defined in terms of the existence of certain normal series.
Let $G$ be a finite $p$-group, where $p$ is an odd prime number, $H$ be a subgroup of $G$ and $θ\in \Irr(H)$ be an irreducible character of $H$. Assume also that $|G:H|=p^2$. Then the character $θ^G$ of $ G$ induced by $θ$ is either a multiple of an irreducible character of $G$, or has at least $\frac{p+1}{2}$ distinct irreducible constituents.
Let G be a finite p-group, where p is a prime number, and $χ$ and $ψ$ be faithful complex irreducible characters of G. We study the relation between the number $η(χ,ψ)$ of distinct irreducible constituents of the product $χψ$ and p.
An element $g$ of a finite group $G$ is said to be vanishing in $G$ if there exists an irreducible character $χ$ of $G$ such that $χ(g)=0$; in this case, $g$ is also called a zero of $G$. The aim of this paper is to obtain structural properties of a factorised group $G=AB$ when we impose some conditions on prime power order elements $g\in A\cup B$ which are (non-)vanishing in $G$.
Let $G$ be a finite group and $p\in π(G)$, and let Irr$(G)$ be the set of all irreducible complex characters of $G$. Let $χ\in {\rm Irr}(G)$, we write ${\rm cod}(χ)=|G:{\rm ker} χ|/χ(1)$, and called it the codegree of the irreducible character $χ$. Let $N\unlhd G$, write ${\rm Irr}(G|N)=\{χ\in {\rm Irr}(G)~|~N\nsubseteq {\rm ker}χ\}$, and ${\rm cod}(G|N)=\{ {\rm cod}(χ) ~|~χ\in{\rm Irr}(G|N)\}.$ In this Ipaper, we prove that if $N\unlhd G$ and every member of ${\rm cod}(G|N')$ is not divisible by some fixed prime $p\in π(G)$, then $N$ has a normal $p$-complement and $N$ is solvable.
Let N be a normal subgroup of a finite group G and consider the set cd(G|N) of degrees of irreducible characters of G whose kernels do not contain N. A number of theorems are proved relating the set cd(G|N) to the structure of N. For example, if N is solvable, its derived length is bounded above by a function of |cd(G|N)|. Also, if |cd(G|N)| is at most 2, then N is solvable and its derived length is at most |cd(G|N)|. If G is solvable and |cd(G|N)| = 3, then the derived length of N is at most 3.
The weights for a finite group G with respect to a prime number p where introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulates a refinement of Alperin's conjecture involving ordinary irreducible characters - with their defect - and, in 2000, Geoffrey Robinson proves that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in…
A finite group $G$ is $normally ~monomial$ if all its irreducible characters are induced from linear characters of normal subgroups of $G$. In this paper, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class. Moreover, we give an upper bound for the largest irreducible character degree of normally monomial $p$-groups of maximal class in terms of $p$.
A method to construct irreducible unitary representations of a hyperspecial compact subgroup of a reductive group over p-adic field with odd p is presented. Our method is based upon Cliffods theory and Weil representations over finite fields. It works under an assumption of the triviality of certain Schur multipliers defined for an algebraic group over a finite field. The assumption of the triviality has good evidences in the case of general linear groups and highly probable in regular cases in general. We will give several examples of classical groups where the Schur multipliers are actually …
In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.
In this note, we formulate an observation that "almost all" irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.
A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant characters for finite reflection groups, nilpotent groups and complete monomial groups. We also propose some conjectures about the structure of the groups admitting such characters.
We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For $SL(r)$ the irreducibility result is the following: let $λ=(a_1\geq a_2\geq ... a_{r-1}\geq 0)$ be the highest weight of an irreducible rational representation $V_λ$ of $SL(r)$. Assume that the integers $a_1+r-1, ~a_2+r-2,..., a_{r-1}+1$ are relatively prime. Then the…
We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.
The object of this short note is to prove a theorem and present a conjecture for the number of even entries in the character table of the symmetric group.
In representation theory of finite groups an important role is played by irreducible characters of p-defect 0, for a prime p dividing the group order. These are exactly those vanishing at the p-singular elements. In this paper we generalize this notion investigating the irreducible characters that are constant at the p-singular elements. We determine all such characters of non-zero defect for alternating, symmetric and sporadic simple groups. We also classify the irreducible characters of quasi-simple groups of Lie type that are constant on the non-identity unipotent elements. In particular, w…
We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.
Given a prime number $p$, every irreducible character $χ$ of a finite group $G$ determines a unique conjugacy class of $p$-subgroups of $G$ which we will call the anchors of $χ$. This invariant has been considered by L. Barker in the context of finite $p$-solvable groups. Besides proving some basic properties of these anchors, we investigate the relation to other $p$-groups which can be attached to irreducible characters, such as defect groups, vertices in the sense of J. A. Green and vertices in the sense of G. Navarro.
This paper computes the irreducible characters of the alternating Hecke algebras, which are deformations of the group algebras of the alternating groups. More precisely, we compute the values of the irreducible characters of the semisimple alternating Hecke algebras on a set of elements indexed by minimal length conjugacy class representatives and we show that these character values determine the irreducible characters completely. As an application we determine a splitting field for the alternating Hecke algebras in the semisimple case.
