Character Degrees Research Articles

The Paradox of Global Multireference Diagnostics.

Modern computational chemistry methods are a useful tool for modeling many chemical systems, but they are challenged by multireference species (e.g., transition metals). A variety of diagnostics have been formulated to identify such cases. They are typically developed by analyzing multireference characters of small molecules, and many provide an average picture of the entire system. We caution the use of such diagnostics for large systems because large systems may include parts with varying degrees of multireference characters. Specifically, a small but highly multireference component may yield a large error in absolute terms, which may be masked in an average value over the entire molecule. As the calculation of molecular relative energies often concerns errors in absolute terms, such a false sense of safety may be detrimental. A prospective means to tackle this challenge is to use fractional occupation density to identify potentially problematic components in a system and then examine this moiety with higher-level computations on appropriately constructed smaller models.

A family of graphs that cannot occur as character degree graphs of solvable groups

A family of graphs that cannot occur as character degree graphs of solvable groups

On p-parts of character degrees

Let G be a finite group and p be a prime. In this paper, we get the sharp bound for |G/Op(G)|p under the assumption that either p2∤χ(1) for all χ∈Irr(G) or p2∤ϕ(1) for all ϕ∈IBrp(G). This would settle two conjectures raised by Lewis, Navarro, Tiep, and Tong-Viet in [7].

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.

On Character Degrees and Codegrees

On Character Degrees and Codegrees

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 .

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.

Even Character Degrees and Ito–Michler Theorem

Even Character Degrees and Ito–Michler Theorem

Two results on character codegrees

Let [Formula: see text] be a finite group and [Formula: see text] be the set of irreducible characters of [Formula: see text]. The codegree of an irreducible character [Formula: see text] of the group [Formula: see text] is defined as [Formula: see text]. In this paper, we study two topics related to the character codegrees. The first result is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms. The second result is about the [Formula: see text]-parts of the codegrees and character degrees.

On $p$-parts of monomial character degrees of solvable groups

On $p$-parts of monomial character degrees of solvable groups

Variations on average character degrees and solvability

Variations on average character degrees and solvability

Supersolvability and nilpotency in terms of the commuting probability and the average character degree

Let p be a prime and let G be a finite group such that the smallest prime that divides |G| is p. We find sharp bounds, depending on p, for the commuting probability and the average character degree to guarantee that G is nilpotent or supersolvable.

Principal blocks with distinct character degrees

Let G be a finite group, p a prime dividing | G | . In this paper, we study the structure of G, which satisfies the condition that any two irreducible characters in its principal p-block have distinct degrees. We provide a reduction of this question to simple groups and determine the structure of G for p = 2. Furthermore, we describe the structure of G in which any two characters of the principal p-block of G have coprime degrees.

Finite Groups Whose Numbers of Real-valued Character Degrees of All Proper Subgroups Are at Most Two

Finite groups with real-valued irreducible characters of prime degree are classified by Dolfi, Pacifici and Sanus. In this paper, the structures of finite groups whose all proper subgroups have at most two real-valued-irreducible-character degrees are determined.

ON p-SOLVABILITY AND AVERAGE CHARACTER DEGREE IN A FINITE GROUP

AbstractAssume that G is a finite group, N is a nontrivial normal subgroup of G and p is an odd prime. Let $\mathrm{Irr}_p(G)=\{\chi \in \mathrm{Irr}(G) : \chi (1)=1~\mathrm{or}~ p \mid \chi (1)\}$ and $\mathrm{Irr}_p(G|N)=\{\chi \in \mathrm{Irr}_p(G) : N \not \leq \mathrm{ker}\,\chi \}$ . The average character degree of irreducible characters of $\mathrm{Irr}_p(G)$ and the average character degree of irreducible characters of $\mathrm{Irr}_p(G|N)$ are denoted by $\mathrm{acd}_p(G)$ and $\mathrm{acd}_p(G|N)$ , respectively. We show that if $\mathrm{Irr}_p(G|N) \neq \emptyset $ and $\mathrm{acd}_p(G|N) < \mathrm{acd}_p(\mathrm{PSL}_2(p))$ , then G is p-solvable and $O^{p'}(G)$ is solvable. We find examples that make this bound best possible. Moreover, we see that if $\mathrm{Irr}_p(G|N) = \emptyset $ , then N is p-solvable and $P \cap N$ and $PN/N$ are abelian for every $P \in \mathrm{Syl}_p(G)$ .

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 .

The degrees of non-monomial characters and solvability of a finite group

For a positive integer l and a finite group G, let , where denotes the set of non-monomial characters of G. In this paper, we prove that if , then G is solvable. In particular, we show that for a non-solvable group G, if , then G = MC is a central product of two subgroups M and C such that and G/M is an M-group. Among the other results, we arrive at the solvability of G by assuming .

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} .