Irreducible Complex Character Research Articles

ON LARGE ORBITS OF FINITE SOLVABLE GROUPS ON CHARACTERS

AbstractWe prove that if a solvable group A acts coprimely on a solvable group G, then A has a relatively ‘large’ orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This improves an earlier result of Keller and Yang [‘Orbits of finite solvable groups on characters’, Israel J. Math.199 (2014), 933–940].

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

Let G be a finite group, and let $$\textrm{cd}(G)$$ denote the set of degrees of the irreducible complex characters of G. Define then the character degree graph $$\Delta (G)$$ as 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 continues the work, started in [8], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph. While, in [8], groups with no composition factors isomorphic to $$\textrm{PSL}_{2}(t^a)$$ (for any prime power $$t^a\ge 4$$ ) were treated, here we consider the complementary situation in the case when $$t$$ is odd and $$t^a> 5$$ . The proof of this classification will be then completed in the third and last paper of this series [7] that deals with the case $$t=2$$ .

Huppert’s conjecture and almost simple groups

Let G be a finite group and \mathsf{cd}(G) 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 whose socle is H_{0}=\mathrm{PSL}(2,q) with q=2^{f} ( f prime) such that \mathsf{cd}(G) =\mathsf{cd}(H) , then there exists an abelian subgroup A of G such that G/A is isomorphic to H . In view of Huppert's conjecture (2000), the main result of this paper gives rise to some examples that G is not necessarily a direct product of A and H , and consequently, we cannot extend this conjecture to almost simple groups.

Characterizable and V-characterizable groups

Let G be a finite group. The spectrum πe(G) is the set of all element orders of G. A vanishing element of G is an element g∈G such that χ(g)= 0 for some irreducible complex character χ of G. Denote by Vo(G) the set of the orders of vanishing elements of G. For a set Ω of positive integers, let h(Ω) (v(Ω)) be the number of isomorphism classes of finite group G such that πe(G)= Ω Vo(G)= Ω, respectively). A group G is called characterizable (V-characterizable) if h(πe(G))= 1 (v(Vo(G))= 1, respectively). In this note, we discuss the relation between characterizable and V-recognizable. Moreover, by an application of the the relation, we prove that the group M22 is V-characterizable.

Regular character-graphs whose eigenvalues are greater than or equal to −2

Let G be a finite group and Irr(G) be the set of all complex irreducible characters of G. The character-graph Δ(G) associated to G, is a graph whose vertex set is the set of primes which divide the degrees of some characters in Irr(G) and two distinct primes p and q are adjacent in Δ(G) if the product pq divides χ(1), for some χ∈Irr(G). Tong-Viet posed the conjecture that if Δ(G) is k-regular for some integer k⩾2, then Δ(G) is either a complete graph or a cocktail party graph. In this paper, we show that his conjecture is true for all regular character-graphs whose eigenvalues are in the interval [−2,∞).

Groups with the same character degrees as almost simple Suzuki groups

Let [Formula: see text] be a finite group and [Formula: see text] denote the set of complex irreducible character degrees of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group and [Formula: see text] is an almost simple group with socle [Formula: see text] such that [Formula: see text], then [Formula: see text] and [Formula: see text] is isomorphic to [Formula: see text].

Characterization of some almost simple groups with socle PSp4(q) by their character degrees

Let [Formula: see text] be a finite group and [Formula: see text] denote the set of irreducible complex character degrees of [Formula: see text]. Suppose that [Formula: see text] is a projective conformal symplectic group [Formula: see text] for some prime power [Formula: see text]. The main result of this paper is to prove that if [Formula: see text], then [Formula: see text] is isomorphic to [Formula: see text]. This confirms a recent generalization of Huppert’s conjecture for the almost simple groups of Lie type.

N-Exact character graphs

Let Γ be a finite simple graph. If for some integer Γ is a Kn -free graph whose complement has an odd cycle of length at least then we say that Γ is an n-exact graph. For a finite group G, let denote the character graph built on the set of degrees of the irreducible complex characters of G. In this paper, we prove that the order of an n-exact character graph is at most Also we determine the structure of all finite groups G with extremal n-exact character graph.

A new characterization of the automorphism groups of Mathieu groups

Abstract Let cd ( G ) {\rm{cd}}\left(G) be the set of irreducible complex character degrees of a finite group G G . ρ ( G ) \rho \left(G) denotes the set of primes dividing degrees in cd ( G ) {\rm{cd}}\left(G) . For any prime p, let p e p ( G ) = max { χ ( 1 ) p ∣ χ ∈ Irr ( G ) } {p}^{{e}_{p}\left(G)}=\max \left\{\chi {\left(1)}_{p}\hspace{0.08em}| \hspace{0.08em}\chi \in {\rm{Irr}}\left(G)\right\} and V ( G ) = { p e p ( G ) ∣ p ∈ ρ ( G ) } V\left(G)=\left\{{p}^{{e}_{p}\left(G)}\hspace{0.08em}| \hspace{0.1em}p\in \rho \left(G)\right\} . The degree prime-power graph Γ ( G ) \Gamma \left(G) of G G is a graph whose vertices set is V ( G ) V\left(G) , and two vertices x , y ∈ V ( G ) x,y\in V\left(G) are joined by an edge if and only if there exists m ∈ cd ( G ) m\in {\rm{cd}}\left(G) such that x y ∣ m xy| m . It is an interesting and difficult problem to determine the structure of a finite group by using its degree prime-power graphs. Qin proved that all Mathieu groups can be uniquely determined by their orders and degree prime-power graphs. In this article, we continue this topic and successfully characterize all the automorphism groups of Mathieu groups by using their orders and degree prime-power graphs.

Notes on normal p-complements and monomial characters

Let G be a finite group and Irr(G) be the set of irreducible (complex) characters of G. Let χ ∈ Irr(G) and write cod(χ) = |G : ker χ|/χ(1) as the codegree of χ, where ker χ is the kernel of χ. Denote by the intersection of the kernels of all the monolithic, monomial irreducible characters of a p-solvable group G with codegrees divisible by a prime p. We prove in this note that has a normal p-complement, and then the theorem in [2] is generalized. Also, we prove that about the theorem in [2] more is true.

On Huppert's rho-sigma conjecture

For an irreducible complex character χ of the finite group G, let π(χ) denote the set of prime divisors of the degree χ(1) of χ. Denote then by ρ(G) the union of all the sets π(χ) and by σ(G) the largest value of |π(χ)|, as χ runs in Irr(G). The ρ-σ conjecture, formulated by Bertram Huppert in the 80's, predicts that |ρ(G)|≤3σ(G) always holds, whereas |ρ(G)|≤2σ(G) holds if G is solvable; moreover, O. Manz and T.R. Wolf proposed a “strengthened” form of the conjecture in the general case, asking whether |ρ(G)|≤2σ(G)+1 is true for every finite group G. In this paper we study the strengthened ρ-σ conjecture for the class of finite groups having a trivial Fitting subgroup: in this context, we prove that the conjecture is true provided σ(G)≤5, but it is false in general if σ(G)≥6. Instead, we establish that |ρ(G)|≤3σ(G)−4 holds for every finite group with a trivial Fitting subgroup and with σ(G)≥6 (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have |ρ(G)|≤3σ(G) whenever G belongs to one particular class including all the finite solvable groups, and we improve the up-to-date best bound obtained in [18] for the general case.

A McKay bijection for projectors

If \mathfrak F is a saturated formation of groups, we define a canonical subset \text{Irr}_{\mathfrak F'}(G) of the irreducible complex characters of a finite solvable group G . If H is an \mathfrak F -projector of G , we show that |\text{Irr}_{\mathfrak F'}(G)|=|\text{Irr}(\mathbf{N}_G(H)/H')| , where H'=[H,H] is the derived subgroup of H . In particular, if \mathfrak F is the class of p -groups, this reproves the solvable case of the celebrated McKay conjecture.

Groups in which the co-degrees of the irreducible characters are distinct

Let G be a finite group and let be the set of all irreducible complex characters of G. For a character the number is called the co-degree of χ. The set of co-degrees of all irreducible characters of G is denoted by In this paper, we show that for a nontrivial finite group G, if and only if G is isomorphic to or S 3.

Two results on the character degree sums

Let G be a finite group, and T(G) be the sum of all complex irreducible character degrees of G. In this paper, we aim to characterize the structure of finite groups in terms of T(G). We show that if |G|/T(G)<(p+1)/2, then G has a normal Sylow p-subgroup; and that if |G|/T(G)<3p2/(p2+2), then G is p-supersolvable, where p is a prime.

Planar character-graphs

For a finite group G, let R(G) be the solvable radical of G. The character-graph of G is a graph whose vertices are the primes which divide the degrees of some irreducible complex characters of G and two distinct primes p and q are joined by an edge if the product pq divides some character degree of G. In this paper we prove that, if has no subgraph isomorphic to and it’s complement is non-bipartite, then is an almost simple group with socle isomorphic to where is a prime power. Also we study the structure of all planar graphs that occur as the character-graph of a finite group G.

Disconnected character graphs and odd dominating sets

Suppose Γ is a finite simple graph. If D is a dominating set of Γ such that each is contained in the set of vertices of an odd cycle of Γ, then we say that D is an odd dominating set for Γ. For a finite group G, let denote the character graph built on the set of degrees of the irreducible complex characters of G. In this paper, we show that the complement of contains an odd dominating set, if and only if is a disconnected graph with non-bipartite complement.

Groups with the same character degrees as sporadic quasisimple groups

Let G be a finite group and cd(G) denote the set of complex irreducible character degrees of G. We prove that if is a sporadic quasisimple group that then there exists an Abelian subgroup A of G such that G is isomorphic to a central product of H and A.

AN ANALOGUE OF HUPPERT’S CONJECTURE FOR CHARACTER CODEGREES

AbstractLet G be a finite group, let ${\mathrm{Irr}}(G)$ be the set of all irreducible complex characters of G and let $\chi \in {\mathrm{Irr}}(G)$ . Define the codegrees, ${\mathrm{cod}}(\chi ) = |G: {\mathrm{ker}}\chi |/\chi (1)$ and ${\mathrm{cod}}(G) = \{{\mathrm{cod}}(\chi ) \mid \chi \in {\mathrm{Irr}}(G)\} $ . We show that the simple group ${\mathrm{PSL}}(2,q)$ , for a prime power $q&gt;3$ , is uniquely determined by the set of its codegrees.

On the singular value decomposition over finite fields and orbits of [formula omitted

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GUm(q)×GUn(q) on Mm×n(q2) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form AA∗ and A∗A over a finite field and AA⊤ and A⊤A over arbitrary fields.

Non-solvable groups having five irreducible character degrees

Let G be a finite group and let be the set of degrees of complex irreducible characters of G. An important program in character theory is to obtain the structure of G when is known. In this article, we consider a dual version of this approach. Indeed, we consider non-solvable groups with five character degrees and present a description of the set of character degrees for such groups. As an application of this result, it is shown that a finite group with five character degrees and containing no prime power must be a solvable group.