Irreducible Character Degrees Research Articles

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.

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.

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 the degrees of irreducible characters fixed by some field automorphism in 𝑝-solvable groups

It is known that, if all the real-valued irreducible characters of a finite group have odd degree, then the group has normal Sylow 2 2 -subgroup. This result is generalized for Sylow p p -subgroups, for any prime number p p , while assuming the group to be p p -solvable. In particular, it is proved that a p p -solvable group has a normal Sylow p p -subgroup if p p does not divide the degree of any irreducible character of the group fixed by a field automorphism of order p p .

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.

Recognition of the symplectic simple group $ PSp_4(p) $ by the order and degree prime-power graph

<abstract><p>Let $ G $ be a finite group, $ \operatorname{cd}(G) $ the set of all irreducible character degrees of $ G $, and $ \rho(G) $ the set of all prime divisors of integers in $ \operatorname{cd}(G) $. For a prime $ p $ and a positive integer $ n $, let $ n_p $ denote the $ p $-part of $ n $. The degree prime-power graph of $ G $ is a graph whose vertex set is $ V(G) = \left\{p^{e_p(G)} \mid p \in \rho(G)\right\} $, where $ p^{e_p(G)} = \max \left\{n_p \mid n \in \operatorname{cd}(G)\right\} $, and there is an edge between distinct numbers $ x, y \in V(G) $ if $ x y $ divides some integer in $ \operatorname{cd}(G) $. The authors have previously shown that some non-abelian simple groups can be uniquely determined by their orders and degree prime-power graphs. In this paper, the authors build on this work and demonstrate that the symplectic simple group $ PSp_4(p) $ can be uniquely identified by its order and degree prime-power graph.</p></abstract>

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.

Representations and Tensor Product Growth

Abstract The deep theory of approximate subgroups establishes three-step product growth for subsets of finite simple groups $G$ of Lie type of bounded rank. In this paper, we obtain two-step growth results for representations of such groups $G$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $G$ be a finite simple group of Lie type and $\chi $ a character of $G$. Let $|\chi |$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $G$ that are constituents of $\chi $. We show that for all $\delta>0$, there exists $\epsilon>0$, independent of $G$, such that if $\chi $ is an irreducible character of $G$ satisfying $|\chi | \le |G|^{1-\delta }$, then $|\chi ^2| \ge |\chi |^{1+\epsilon }$. We also obtain results for reducible characters and establish faster growth in the case where $|\chi | \le |G|^{\delta }$. In another direction, we explore covering phenomena, namely situations where every irreducible character of $G$ occurs as a constituent of certain products of characters. For example, we prove that if $|\chi _1| \cdots |\chi _m|$ is a high enough power of $|G|$, then every irreducible character of $G$ appears in $\chi _1\cdots \chi _m$. Finally, we obtain growth results for compact semisimple Lie groups.

Counterexamples to a conjecture about the sum of degrees of irreducible characters

Counterexamples to a conjecture about the sum of degrees of irreducible characters

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

For a positive integer n and a prime p, let np denote the p-part of n. Let G be a group, the set of all irreducible character degrees of G, the set of all prime divisors of integers in where The authors proved that if and only if and In this paper, the authors continue this topic and prove that if and only if and

Finite non-solvable groups whose real degrees are prime-powers

Abstract We present a description of non-solvable groups in which all real irreducible character degrees are prime-power numbers.

Finite groups for which all proper subgroups have consecutive character degrees

<abstract><p>Huppert and Qian et al. classified finite groups for which all irreducible character degrees are consecutive. The aim of this paper is to determine the structure of finite groups whose irreducible character degrees of their proper subgroups are all consecutive.</p></abstract>

Character degrees in blocks and defect groups

A recent question of Gabriel Navarro asks whether it is true that the derived length of a defect group is less than or equal to the number of degrees of irreducible characters in a block. In this article, we bring new evidence towards the validity of this statement.

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.

A supercharacter theory based on the degrees of irreducible characters

A supercharacter theory based on the degrees of irreducible characters

Nonsolvable groups whose irreducible character degrees have special 2-parts

Nonsolvable groups whose irreducible character degrees have special 2-parts

On Groups Whose Irreducible Character Degrees of All Proper Subgroups are All Prime Powers

Isaacs, Passman, and Manz have determined the structure of finite groups whose each degree of the irreducible characters is a prime power. In particular, if G is a nonsolvable group and every character degree of a group G is a prime power, then G is isomorphic to S × A , where S ∈ A 5 , PSL 2 8 and A is abelian. In this paper, we change the condition, each character degree of a group G is a prime power, into the condition, each character degree of the proper subgroups of a group is a prime power, and give the structure of almost simple groups whose character degrees of all proper subgroups are all prime powers.