Degree Graphs of Simple Groups of Exceptional Lie Type

Abstract

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type.     

Degree Graphs of Simple Linear and Unitary Groups

Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ? cd (G). We determine the graph Δ(G) for the finite simple groups of types A _?(q) and ² A _? (q ²), that is, for the simple linear and unitary groups.     

A graph associated with the p\pi-character degrees of a group

Let G be a group and $\pi$ be a set of primes. We consider the set ${\rm cd}^{\pi}(G)$ of character degrees of G that are divisible only by primes in $\pi$. In particular, we define $\Gamma^{\pi}(G)$ to be the graph whose vertex set is the set of primes dividing degrees in ${\rm cd}^{\pi}(G)$. There is an edge between p and q if pq divides a degree $a \in {\rm cd}^{\pi}(G)$. We show that if G is $\pi$-solvable, then $\Gamma^{\pi}(G)$ has at most two connected components.     

Bounding the Derived Length of a Solvable Group: An Improvement on a Result by Gluck

Let cd(G) be the set of degrees of irreducible complex characters and dl(G) be the derived length of the finite group G. It is a result by Gluck that if G is solvable then dl(G) ≤2|cd(G)|. Using a result of Isaacs and Knutson, I will in this article show that this bound can be improved to dl(G) ≤2|cd(G)| ?3 when |cd(G)| ≥3.     

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.     

Simple exceptional groups of Lie type are determined by their character degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G)={c(1)  |  c ? Irr(G)}{{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}} be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and cd(S) í cd(H){{\rm cd}(S)\subseteq {\rm cd}(H)} then S must be isomorphic to H. As a consequence, we show that if G is a finite group with X₁(G) í X₁(H){{\rm X}_1(G)\subseteq {\rm X}_1(H)} then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.     

Verifying Huppert's Conjecture for PSL3(q) and PSU3(q 2)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ? H × A, where A is an abelian group. We examine arguments to verify this conjecture for the simple groups of Lie type of rank two. To illustrate our arguments, we extend Huppert's results and verify the conjecture for the simple linear and unitary groups of rank two.     

SOLVABLE GROUPS WITH CHARACTER DEGREE GRAPHS HAVING 5 VERTICES AND DIAMETER 3

ABSTRACT

Let G be a finite group and cd(G) the character degrees of G. The degree graph Δ(G) of G is the graph whose vertices are the primes dividing degrees in cd(G), and there is an edge between p and q if pq divides some degree in cd(G). In this paper, we show that if Δ(G) has 5 vertices, then the diameter of Δ(G) is at most 2. This shows that the example in^[9] of a solvable group G where Δ(G) has diameter 3 has the fewest number of vertices possible.     

Taketa’s theorem for some character degree sets

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then ${{\rm dl}(G) \leqslant |{\rm cd} (G)|}$ , where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality ${{\rm dl}(N) \leqslant |{\rm cd} {(G \mid N)}|}$ holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if ${{\rm cd} {(G \mid N')}}$ is a set of non-trivial p–powers for some fixed prime p.     

3-connected graphs with non-cut contractible edge covers of size k

In this paper, we show that if a 3-connected graph G other than K₄ has a vertex subset K that covers the set of contractible edges of G and if |K| 3 and |V(G)| 3|K| ? 1, then K is a cutset of G. We also give examples to show that this result is best possible. In particular, the result does not hold for K with smaller cardinality.     

The size of graphs without nowhere-zero 4-flows

Let G be a 2-edge-connected simple graph with order n. We show that if | V(G)| ≤ 17, then either G has a nowhere-zero 4-flow, or G is contractible to the Petersen graph. We also show that for n large, if | V(G)| n ? 17/2 + 34, then either G has a nonwhere-zero 4-flow, or G can be contracted to the Petersen graph. © 1995 John Wiley & Sons, Inc.     

Prime Divisibility Among Degrees of Solvable Groups

Abstract

Let G be a finite, nonabelian, solvable group. Following work by D. Benjamin, we conjecture that some prime must divide at least a third of the irreducible character degrees of G. Benjamin was able to show the conjecture is true if all primes divide at most 3 degrees. We extend this result by showing if primes divide at most 4 degrees, then G has at most 12 degrees. We also present an example showing our result is best possible.     

The security number of lexicographic products

A subset S of vertices of a graph G is a secure set if |N [X] ∩ S| ≥ |N [X] ? S| holds for any subset X of S, where N [X] denotes the closed neighborhood of X. The minimum cardinality s(G) of a secure set in G is called the security number of G. We investigate the security number of lexicographic product graphs by defining a new concept of tightly-securable graphs. In particular we derive several exact results for different families of graphs which yield some general results.     

Derived Lengths of Solvable Groups Having Five Irreducible Character Degrees I

Let G be a solvable group with five character degrees. We show that the derived length of G is at most 5. This verifies that the Taketa inequality, dl(G)|cd(G)|, is valid for solvable groups with at most five character degrees.     

An answer to a question of Isaacs on character degree graphs

Let N be a normal subgroup of a finite group G. We consider the graph Γ(G|N) whose vertices are the prime divisors of the degrees of the irreducible characters of G whose kernel does not contain N and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of G whose kernel does not contain N. We prove that if Γ(G|N) is disconnected then G/N is solvable. This proves a strong form of a conjecture of Isaacs.     

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 get the exact lower bound of |G|∕T(G) for a non-r-solvable group G.     

Common Divisor Character Degree Graphs of Solvable Groups with Four Vertices

Let G be a finite solvable group. The common divisor graph Γ(G) attached to G is a character degree graph. Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor (m, n) > 1. In this article, we classify all graphs with four vertices that may occur as Γ(G) for solvable group G.     

Balanced coloring of bipartite graphs

Given a bipartite graph G(U∪V, E) with n vertices on each side, an independent set I∈G such that |U∩I|=|V∩I| is called a balanced bipartite independent set. A balanced coloring of G is a coloring of the vertices of G such that each color class induces a balanced bipartite independent set in G. If graph G has a balanced coloring we call it colorable. The coloring number χ_B(G) is the minimum number of colors in a balanced coloring of a colorable graph G. We shall give bounds on χ_B(G) in terms of the average degree $\bar{d}$ of G and in terms of the maximum degree Δ of G. In particular we prove the following:

• $\chi_{{{B}}}({{G}}) \leq {{max}} \{{{2}},\lfloor {{2}}\overline{{{d}}}\rfloor+{{1}}\}$.
• For any 0<ε<1 there is a constant Δ₀ such that the following holds. Let G be a balanced bipartite graph with maximum degree Δ≥Δ₀ and n≥(1+ε)2Δ vertices on each side, then $\chi_{{{B}}}({{G}})\leq \frac{{{{20}}}}{\epsilon^{{{2}}}} \frac{\Delta}{{{{ln}}}\,\Delta}$.

© 2009 Wiley Periodicals, Inc. J Graph Theory 64: 277–291, 2010     

Existence ofN-INJECTORS in a not central normal Fitting class

If we denote byL(G) the semisimple radical of a groupG, we prove in this paper that ℒ=G|G=C_G(L(G))L(G) is a not central normal Fitting class. Moreover, all ( haveN-groups haveN-injectors.