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.     

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.     

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.     

On the Diameter of a p-Regular Conjugacy Class Graph of Finite Groups II

Abstract

Let Gbe a finite p-solvable group. Let us consider the graph Γ* _p (G) whose vertices are the primes which occur as the divisors of the conjugacy classes of p-regular elements of G and two primes are joined by an edge if there exists such a class whose size is divisible by both primes. Suppose that Γ _p *(G) is a connected graph, then we prove that the diameter of this graph is at most 3 and this is the best bound.     

An algorithm and self-reproducing systems

Let G be a group and let N be a normal subgroup of G. We set cd(G|N) to be the degrees of the irreducible characters of G whose kernels do not contain N. We associate a graph with this set. The vertices of this graph are the primes dividing degrees in cd(G|N), and there is an edge between p and q if pq divides some degree in cd(G|N). In this paper, we study this graph when it is disconnected, and we study its diameter when it is connected.     

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.     

Lamron ℓ-groups

The article introduces a new class of lattice-ordered groups. An ?-group G is lamron if Min(G)^?1 is a Hausdorff topological space, where Min(G)^?1 is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron ?-groups are related to ?-groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron ?-group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented ?-groups.     

The longest cycle of a graph with a large minimal degree

We show that every graph G on n vertices with minimal degree at least n/k contains a cycle of length at least [n/(k ? 1)]. This verifies a conjecture of Katchalski. When k = 2 our result reduces to the classical theorem of Dirac that asserts that if all degrees are at least 1/2n then G is Hamiltonian.     

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.     

Maximal cyclic subgroups and prime divisors in finite groups

Let G be a finite group with no chief factor simple of Lie type E ₈(q) and C a cyclic subgroup of largest order in G. It is shown that at most two primes in the open interval ([|C|/2], |C|) divide |G|.Received: 14 January 2005     

The McKay conjecture for exceptional groups and odd primes

Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G → G and G := G ^F , such that the root system is of exceptional type or G is a Suzuki group or Steinberg’s triality group. We show that all irreducible characters of C _G (S), the centraliser of S in G, extend to their inertia group in N _G (S), where S is any F-stable Sylow torus of (G, F). Together with the work in [16] this implies that the McKay conjecture is true for G and odd primes ℓ different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [14]. This research has been supported by the DFG-grant “Die Alperin-McKay-Vermutung für endliche Gruppen” and an Oberwolfach Leibniz fellowship.     

A graph-theoretic version of the union-closed sets conjecture

An induced subgraph S of a graph G is called a derived subgraph of G if S contains no isolated vertices. An edge e of G is said to be residual if e occurs in more than half of the derived subgraphs of G. We introduce the conjecture: Every non-empty graph contains a non-residual edge. This conjecture is implied by, but weaker than, the union-closed sets conjecture. We prove that a graph G of order n satisfies this conjecture whenever G satisfies any one of the conditions: δ(G) ≤ 2, log₂ n ≤ δ(G), n ≤ 10, or the girth of G is at least 6. Finally, we show that the union-closed sets conjecture, in its full generality, is equivalent to a similar conjecture about hypergraphs. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 155–163, 1997     

Average distance and maximum induced forest

With the help of the Graffiti system, Fajtlowicz conjectured around 1992 that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced bipartite subgraph of G, denoted α₂(G). We prove a strengthening of this conjecture by showing that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced forest, denoted F(G). Moreover, we characterize the graphs maximizing the average distance among all graphs G having a fixed number of vertices and a fixed value of F(G) or α₂(G). Finally, we conjecture that the average distance between two vertices of a connected graph is at most half the maximum order of an induced linear forest (where a linear forest is a union of paths). © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 31–54, 2009     

Decomposing Oriented Graphs into Six Locally Irregular Oriented Graphs

An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition \(E_1 \cup E_2 \cup \cdots \cup E_k\) of the edge set E(G) such that each \(E_i\) induces a locally irregular graph. It was recently conjectured by Baudon et al. that every undirected graph admits a decomposition into at most three locally irregular graphs, except for a well-characterized set of indecomposable graphs. We herein consider an oriented version of this conjecture. Namely, can every oriented graph be decomposed into at most three locally irregular oriented graphs, i.e. whose adjacent vertices have distinct outdegrees? We start by supporting this conjecture by verifying it for several classes of oriented graphs. We then prove a weaker version of this conjecture. Namely, we prove that every oriented graph can be decomposed into at most six locally irregular oriented graphs. We finally prove that even if our conjecture were true, it would remain NP-complete to decide whether an oriented graph is decomposable into at most two locally irregular oriented graphs.     

On the distribution of the number of customers in the symmetric M/G/1 queue

We consider an M/G/1 queue with symmetric service discipline. The class of symmetric service disciplines contains, in particular, the preemptive last-come-first-served discipline and the processor-sharing discipline. It has been conjectured in Kella et al. [1] that the marginal distribution of the queue length at any time is identical for all symmetric disciplines if the queue starts empty. In this paper we show that this conjecture is true if service requirements have an Erlang distribution. We also show by a counterexample, involving the hyperexponential distribution, that the conjecture is generally not true. AMS Subject Classifications Primary—60K25; Secondary—90B22     

Strongly perfect infinite graphs

An infinite graph G is calledstrongly perfect if each induced subgraph ofG has a coloring (C _i :i ∈I) and a clique meeting each colorC _i . A conjecture of the first author and V. Korman is that a perfect graph with no infinite independent set is strongly perfect. We prove the conjecture for chordal graphs and for their complements. The research was begun at the Sonderforschungsbereich 343 at Bielefeld University and supported by the Fund for the Promotion of Research at the Technion.     

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.     

On 3-colorings of Plane Graphs

In this paper, we prove that if G is a plane graph without 4-, 5- and 7-circuits and without intersecting triangles, then for each face f of degree at most 11, any 3-coloring of the boundary of f can be extended to G. This gives a positive support to a conjecture of Borodin and Raspaud which claims that each plane graph without 5-circuits and intersecting triangles is 3-colorable.     

On monomial representations of finitely generated nilpotent groups

A result of Segal states that every complex irreducible representation of a finitely generated nilpotent group G is monomial if and only if G is abelian-by-finite. A conjecture of Parshin, recently proved affirmatively by Beloshapka and Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture using ideas of Kutzko and Brown. We also give a characterization of the finite-dimensional irreducible representations of two-step nilpotent groups and describe these completely for two-step groups whose center has rank one.     

FINITELY GENERATED G′ AND POSITIVE LAWS

We conjecture that a finitely generated relatively free group G has a finitely generated commutator subgroup G′ if and only if G satisfies a positive law. We confirm this conjecture for groups G in the large class, containing all residually finite and all soluble groups.