Irreducible Complex Character Research Articles

On Gluck’s conjecture for wreath product type groups

Abstract A well-known conjecture of Gluck claims that | G : F ( G ) | ≤ b ( G ) 2 \lvert G:\mathbf{F}(G)\rvert\leq b(G)^{2} for all finite solvable groups 𝐺, where F ⁢ ( G ) \mathbf{F}(G) is the Fitting subgroup and b ⁢ ( G ) b(G) is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form G ≀ H 1 ≀ H 2 ≀ ⋯ ≀ H r G\wr H_{1}\wr H_{2}\wr\cdots\wr H_{r} , where 𝐺 is a finite solvable group acting primitively on F ⁢ ( G ) / Φ ⁢ ( G ) \mathbf{F}(G)/\Phi(G) , and each H i H_{i} is a solvable primitive permutation group of finite degree.

Some Simple Groups Which are Determined by Their Orders and Degree-Patterns

Let [Formula: see text] be a finite group and [Formula: see text] the set of all irreducible complex characters of [Formula: see text]. Let [Formula: see text] be the set of all irreducible complex character degrees of [Formula: see text] and denote by [Formula: see text] the set of all primes which divide some character degree of [Formula: see text]. The character-prime graph [Formula: see text] associated to [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] and there is an edge between two distinct primes [Formula: see text] and [Formula: see text] if and only if the product [Formula: see text] divides some character degree of [Formula: see text]. We show that the finite nonabelian simple groups [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are uniquely determined by their degree-patterns and orders.

Primes and conductors of characters

If G is a finite p-solvable group, P is a Sylow p-subgroup of G and χ∈Irr(G) is an irreducible complex character of degree not divisible by p, we prove that the field of values Q(χP) of the restriction of χ to P has index not divisible by p in the cyclotomic extension Q(e2πipf), where pf is the p-part of the conductor of χ. This proves the p-solvable case of the main conjecture in [4].

Characters of [formula omitted] and vertex operators

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group GLn(Fq). Green's theory of GLn(Fq) is recovered and enhanced under the realization of the Grothendieck ring of representations RG=⨁n≥0R(GLn(Fq)) as two isomorphic Fock spaces associated to two infinite-dimensional F-equivariant Heisenberg Lie algebras hˆF‾ˆq and hˆF‾q, where F is the Frobenius automorphism of the algebraically closed field F‾q. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space RG as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example.

P-Singular characters and normal Sylow p-subgroups

Let [Formula: see text] be a finite group and [Formula: see text] a prime. We denote by [Formula: see text] the set of irreducible complex characters of [Formula: see text] whose degrees are linear or divisible by [Formula: see text], and we write [Formula: see text] to denote the ratio of the sum of squares of irreducible character degrees in [Formula: see text] to the sum of irreducible character degrees in [Formula: see text]. The Itô–Michler Theorem on character degrees states that [Formula: see text] if and only if [Formula: see text] has a normal abelian Sylow [Formula: see text]-subgroup. We generalize this theorem as follows: if [Formula: see text], then [Formula: see text] has a normal Sylow [Formula: see text]-subgroup.

Eulerian character degree graphs of solvable groups

Let G be a finite group, let Irr(G) be the set of all complex irreducible characters of G and let cd(G) be the set of all degrees of characters in Irr ( G ) . Let ρ ( G ) be the set of all primes that divide some degrees in cd ( G ) . The character degree graph Δ ( G ) of G is the simple undirected graph with vertex set ρ ( G ) and in which two distinct vertices p and q are adjacent if there exists a character degree r ∈ cd ( G ) such that r is divisible by the product pq. In this paper we obtain a necessary condition for the character degree graph Δ ( G ) of a solvable group G to be eulerian. We also prove that every n – 2 regular graph on n vertices where n is even and n ≥ 4 is the character degree graph of some solvable group. In the last part of the paper we give a bound for the number of Eulerian character degree graphs in terms of number of vertices.

The characterization of PSL(2, p) (p≡±5 (mod 24))

Let G be a finite group. A vanishing element of G is an element g ∈ G such that χ ( g ) = 0 for some irreducible complex character χ of G. We will denote by Van(G) the set of vanishing elements of G and by Vo ( G ) the set of the order of elements in Van ( G ) . For certain specified primes p that satisfy p ≡ ± 5 ( mod 24 ) , if Vo ( G ) = Vo ( PSL ( 2 , p ) ) , then G ≅ PSL ( 2 , p ) .

Element orders and codegrees of characters in non-solvable groups

Given a finite group G and an irreducible complex character χ of G, the codegree of χ is defined as the integer cod(χ)=|G:ker(χ)|/χ(1). It was conjectured by G. Qian in [16] that, for every element g of G, there exists an irreducible character χ of G such that cod(χ) is a multiple of the order of g; the conjecture has been verified under the assumption that G is solvable ([16]) or almost-simple ([13]). In this paper, we prove that Qian's conjecture is true for every finite group whose Fitting subgroup is trivial, and we show that the analysis of the full conjecture can be reduced to groups having a solvable socle.

Galois Conjugacy and Fitting Height 2: Classification of Finite Solvable Groups

This paper investigates the finite groups 𝐺 for which any two characters in the set of irreducible complex characters 𝑰𝒓𝒓(𝐺) are Galois conjugate. Specifically, we classify such groups and establish a key result: they are solvable with Fitting height 2. The analysis involves intricate considerations of irreducible complex characters and their Galois conjugacy, shedding light on the structural properties of finite solvable groups.

Action of automorphisms on irreducible characters of finite reductive groups of type 𝖠

Abstract Let 𝐺 be a finite reductive group such that the derived subgroup of the underlying algebraic group is a product of quasi-simple groups of type 𝖠. In this paper, we give an explicit description of the action of automorphisms of 𝐺 on the set of its irreducible complex characters. This generalizes a recent result of M. Cabanes and B. Späth [Equivariant character correspondences and inductive McKay condition for type 𝖠, J. Reine Angew. Math. 728 (2017), 153–194] and provides a useful tool for investigating the local sides of the local-global conjectures as one usually needs to deal with Levi subgroups. As an application we obtain a generalization of the stabilizer condition in the so-called inductive McKay condition [B. Späth, Inductive McKay condition in defining characteristic, Bull. Lond. Math. Soc. 44 (2012), 3, 426–438; Theorem 2.12] for irreducible characters of 𝐺. Moreover, a criterion is given to explicitly determine whether an irreducible character is a constituent of a given generalized Gelfand–Graev character of 𝐺.

On group actions of p‐groups on p′$p^{\prime }$‐groups

AbstractLet A be a finite group that acts coprimely on another finite group G, and in this setting we study the orbit structure of the group A on the irreducible characters or conjugacy classes of the group G. We prove that if a p‐group A acts coprimely on a finite group G, then A has a “large” orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This answers a conjecture of Moretó affirmatively.

On the product of vanishing classes in a finite group

Let G be a finite group. An element g ∈ G is called a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0. The conjugacy class of G containing a vanishing element is called a vanishing class of G. In this paper, we concern on the finite groups G such that the product of every two non-inverse vanishing classes of G is a conjugacy class of G. Also, we study the finite groups G such that product of every two vanishing classes of G which are not inverse modulo the center of G is a conjugacy class of G.

A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters

Abstract Let 𝐺 be a finite group and let Irr s ⁢ ( G ) \mathrm{Irr}_{\mathfrak{s}}(G) be the set of irreducible complex characters 𝜒 of 𝐺 such that χ ⁢ ( 1 ) 2 \chi(1)^{2} does not divide the index of the kernel of 𝜒. In this paper, we classify the finite groups 𝐺 for which any two characters in Irr s ⁢ ( G ) \mathrm{Irr}_{\mathfrak{s}}(G) are Galois conjugate. In particular, we show that such groups are solvable with Fitting height 2.

Finite groups in which distinct nonlinear irreducible characters have distinct codegrees

AbstractFor an irreducible complex character χ of a finite group G, its codegree is defined as . The main focus of this paper is to provide a classification of finite groups where distinct nonlinear irreducible characters have distinct codegrees.

Non-solvable Groups whose Character Degree Graph has a Cut-Vertex. I

Let G be a finite group. Denoting by textrm{cd}(G) the set of degrees of the irreducible complex characters of G, we consider the character degree graph of G: this is the (simple undirected) graph whose vertices are the prime divisors of the numbers in textrm{cd}(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in textrm{cd}(G). In the series of three papers starting with the present one, we analyze the structure of the finite non-solvable groups whose character degree graph possesses a cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph.

5-Regular prime graphs of finite nonsolvable groups

Abstract The prime graph Δ ⁢ ( G ) \Delta(G) of a finite group 𝐺 is a graph whose vertex set is the set of prime factors of the degrees of all irreducible complex characters of 𝐺, and two distinct primes 𝑝 and 𝑞 are joined by an edge if the product p ⁢ q pq divides some character degree of 𝐺. In 2014, Tong-Viet [H. P. Tong-Viet, Finite groups whose prime graphs are regular, J. Algebra 397 (2014), 18–31] proposed the following conjecture. Let 𝐺 be a group and let k ≥ 5 k\geq 5 be odd. If the prime graph Δ ⁢ ( G ) \Delta(G) is 𝑘-regular, then Δ ⁢ ( G ) \Delta(G) is a complete graph of order k + 1 k+1 . In this paper, we show that if the prime graph Δ ⁢ ( G ) \Delta(G) of a finite nonsolvable group 𝐺 is 5-regular, then Δ ⁢ ( G ) \Delta(G) is isomorphic to the complete graph K 6 K_{6} or possibly the graph depicted in the first figure below. Moreover, if 𝐺 is an almost simple group, then Δ ⁢ ( G ) \Delta(G) is isomorphic to the complete graph K 6 K_{6} .

On groups with the same character degrees as almost simple groups with socle small Ree groups

Let G be a finite group and denote the set of complex irreducible character degrees of G. In this paper, we prove that if G is a finite group and H is an almost simple group with socle , where with odd such that , then G is non-solvable and the chief factor of G is isomorphic to H 0. If, in particular, f is coprime to 3, then is isomorphic to H 0 and is isomorphic to H.

On the largest Brauer and p′-character degrees

Let K be a field of characteristic p>0. We prove that if all irreducible representations over K of a finite group G have degree at most n, then G has a characteristic subgroup N of index bounded above in terms of n such that its derived subgroup N′ is a p-group. Our proof of this result relies on the celebrated Larsen-Pink theorem. We also show that every finite group G has a solvable p-nilpotent subgroup of index bounded above in terms of the largest p′-degree of the complex irreducible characters of G.

Non-solvable groups whose character degree graph has a cut-vertex. III

Let G be a finite group. Denoting by textrm{cd}(G) the set of the degrees of the irreducible complex characters of G, we consider the character degree graph of G: this, is the (simple, undirected) graph whose vertices are the prime divisors of the numbers in textrm{cd}(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in textrm{cd}(G). This paper completes the classification, started in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119), of the finite non-solvable groups whose character degree graph has a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph. More specifically, it was proved in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119 that these groups have a unique non-solvable composition factor S, and that S is isomorphic to a group belonging to a restricted list of non-abelian simple groups. In Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119) all isomorphism types for S were treated, except the case Scong textrm{PSL}_{2}(2^a) for some integer age 2; the remaining case is addressed in the present paper.

The uniqueness of vertex pairs in π-separable groups

Let G be a finite π-separable group, where π is a set of primes, and let χ be an irreducible complex character that is a π-lift of some π-partial character of G. It was proved by Cossey and Lewis that all of the vertex pairs for χ are linear and conjugate in G if , but the result can fail for . In this paper we introduce the notion of the linear twisted vertices in the case where , and then establish the uniqueness for such vertices under the conditions that either χ is an -lift for a π-chain of G or it has a linear Navarro vertex, thus answering a question proposed by them.