Set Of Character Degrees Research Articles

Character degrees of 5-groups of maximal class

Abstract Let 𝐺 be a 5-group of maximal class with major centralizer G 1 = C G ⁢ ( G 2 / G 4 ) G_{1}=C_{G}({G_{2}}/{G_{4}}) . In this paper, we prove that the irreducible character degrees of a 5-group 𝐺 of maximal class are almost determined by the irreducible character degrees of the major centralizer G 1 G_{1} and show that the set of irreducible character degrees of a 5-group of maximal class is either { 1 , 5 , 5 3 } \{1,5,5^{3}\} or { 1 , 5 , … , 5 k } \{1,5,\ldots,5^{k}\} with k ≥ 1 k\geq 1 .

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.

Character degrees of normally monomial 𝑝-groups of maximal class

AbstractA 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 study the largest irreducible character degree and the maximal abelian normal subgroup of normally monomial 𝑝-groups of maximal class in terms of 𝑝. In particular, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class.

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.

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.

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.

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.

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.

An overview on the bipartite divisor graph for the set of irreducible character degrees

Let G be a finite group. The bipartite divisor graph B(G) for the set of irreducible complex character degrees cd(G) is the undirected graph with vertex set consisting of the prime numbers dividing some element of cd(G) and of the nonidentity character degrees in cd(G), where a prime number p is declared to be adjacent to a character degree m if and only if p divides m. The graph B(G) is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees. The scope of this paper is two-fold. We draw some attention to B(G) by outlining the main results that have been proved so far, see for instance [10; 11; 25; 26; 27]. In this process we improve some of these results.

1-Connected character-graphs of finite groups with non-bipartite complement

For a finite group [Formula: see text], let [Formula: see text] be the character-graph which is built on the set of irreducible complex character degrees of [Formula: see text]. In this paper, we wish to determine the structure of finite groups [Formula: see text] such that [Formula: see text] is [Formula: see text]-connected with nonbipartite complement. Also, we classify all [Formula: see text]-connected graphs with nonbipartite complement that can occur as the character-graph [Formula: see text] of a finite group [Formula: see text].

Degrees of irreducible representations of direct products of nonabelian simple groups

For a finite group G, let the character degree set of G, denoted by $$\mathrm{cd}(G)$$, be the set of the degrees of all irreducible complex representations of G. In the present paper, the structure of finite groups whose character degree sets coincide with the character degree sets of direct product of nonabelian simple groups will be studied. For a finite group S and a positive integer n, let $$S^{n}$$ be the direct product of n copies of S. We prove that if G is a finite group with $$\mathrm{cd}(G)=\mathrm{cd}(H)$$, where $$\begin{aligned}&H\in \{\mathrm{PSL}_{3}(q)^{n}(q=2^{\alpha }\geqslant 4, 3\not \mid q-1), \mathrm{PSU}_{3}(q)^{n}(q=2^{\alpha }\geqslant 4, 3\not \mid q+1), \\&\quad M_{11}^{n} (1\leqslant n\leqslant 2), M_{23}^{n}, J_{1}^{n}, J_{2}^{n}, J_{3}^{n}, J_{4}^{n}\}, \end{aligned}$$then G is a quasi perfect group. This extends the first step of Huppert’s Conjecture to the direct product of simple groups. This conjecture states that the nonabelian simple groups are uniquely determined up to an abelian direct factor by the set of character degrees.

Groups where the centers of the irreducible characters form a chain

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.

Some Nonsolvable Character Degree Sets

For positive integer k and nonabelian simple group S, let $$S^{k}$$ be the direct product of k copies of S. We conjecture that all finite groups G with $$\mathrm{cd}(G)=\mathrm{cd}(S^{k})$$ are quasi perfect groups (that is; $$G'=G''$$) and hence nonsolvable groups, where $$\mathrm{cd}(G)$$ is the set of irreducible character degrees of G. In this paper, we prove this conjecture for $$S\in \{\mathrm{PSL}_{2}(p^{f}), \mathrm{PSL}_{2}(2^{f}), \mathrm{Sz}(q)\}$$, where $$p>2$$ is an odd prime number such that $$p^{f}>5$$ and $$p^{f}\pm 1\not \mid 2^{k}$$, and $$q=2^{2n+1}\geqslant 8$$.

Some character degree sets implying nonsolvability

Let cd(G) be the set of the degrees of all complex irreducible characters of a finite group G. For a finite nonabelian simple group S and a positive integer k, let Sk be the direct product of k copies of S. In [2], we conjectured that all finite groups G with cd(G)=cd(Sk) are quasi perfect groups (that is; G′=G″) and hence nonsolvable groups. Then we proved that this conjecture holds for some sporadic simple groups as well as for some simple groups of Lie type (see [1] and [2]). In this paper, we verify this conjecture for some alternating groups and for the simple groups Psp4(q)(q=2m≥2) and G22(q2)(q=32m+1≥27). Indeed, we show that if G is a finite group with cd(G)=cd(H), where H∈{A7k,S7k(k≥1),Psp4(q)k(q=2m≥2,k≥1),G22(q2)k(q=32m+1≥27,1≤k≤6560),A8k(1≤k≤5),S8,A9k,S9k,A10k,S10k(1≤k≤2)}, then G is a quasi perfect group.

‎$‎4‎$‎-Regular prime graphs of nonsolvable groups

Let $G$ be a finite group and $cd(G)$ denote the character degree set for $G$. The prime graph $DG$ is a simple graph whose vertex set consists of prime divisors of elements in $cd(G)$, denoted $rho(G)$. Two primes $p,qin rho(G)$ are adjacent in $DG$ if and only if $pq|a$ for some $ain cd(G)$. We determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group.

Mathieu groups and its degree prime-power graphs

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The degree graph of G related to cd(G) was defined. It was proved many finite simple groups (but not all Mathieu groups) are uniquely determined by their orders and degree graphs. We hope to define a new graph related to cd(G) such that more simple groups can be uniquely determined by their orders and this newly defined graphs. Here a degree prime-power graph is defined and it is proved that all Mathieu groups can be determined uniquely by their orders and degree prime-power graphs.

Isaacs–Seitz conjecture for certain groups

Let G be a finite group. I. M. Isaacs and G. Seitz have conjectured that if G is a solvable group, then Taketa’s inequality holds, where is the set of irreducible character degrees of G and is the derived length of G. In this paper, we show that this inequality holds if G is a solvable Frobenius group. Also, we prove that if for some normal Sylow p-subgroup P of G, then . Moreover, we investigate this conjecture for some sets of the real-valued irreducible character degrees.

Groups whose bipartite divisor graph for character degrees has four or fewer vertices

Let [Formula: see text] be a finite group and [Formula: see text] be the set of nonlinear irreducible character degrees of [Formula: see text]. Suppose that [Formula: see text] is the set of primes dividing some elements of [Formula: see text]. The bipartite divisor graph for [Formula: see text], [Formula: see text], is a graph whose vertices are the disjoint union of [Formula: see text] and [Formula: see text], and a vertex [Formula: see text] is connected to a vertex [Formula: see text] if and only if [Formula: see text]. In this paper, we consider groups whose graph has four or fewer vertices. We show that all these groups are solvable and determine the structure of these groups. We also provide examples of any possible graph.

On regular bipartite divisor graph for the set of irreducible character degrees

Given a finite group $G$, the \textit{bipartite divisor graph}, denoted by $B(G)$, for its irreducible character degrees is the bipartite graph with bipartition consisting of $cd(G)^{*}$, where $cd(G)^{*}$ denotes the nonidentity irreducible character degrees of $G$ and the $\rho(G)$ which is the set of prime numbers that divide these degrees, and with $\{p,n\}$ being an edge if $\gcd(p,n)\neq 1$. In [Bipartite divisor graph for the set of irreducible character degress, Int. J. Group Theory, 2017], the author considered the cases where $B(G)$ is a path or a cycle and discussed some properties of $G$. In particular she proved that $B(G)$ is a cycle if and only if $G$ is solvable and $B(G)$ is either a cycle of length four or six. Inspired by $2$-regularity of cycles, in this paper we consider the case where $B(G)$ is an $n$-regular graph for $n\in\{1,2,3\}$. In particular we prove that there is no solvable group whose bipartite divisor graph is $C_{4}+C_{6}$.