Mark L. Lewis

Character degree graphs that are complete graphs

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define F(G) to be the graph whose vertex set is cd(G) - {1}, and there is an edge between a and b if (a, b) > 1. We prove that if Γ(G) is a complete graph, then G is a solvable group.

ISOMORPHISM IN EXPANDING FAMILIES OF INDISTINGUISHABLE GROUPS

Abstract. For every odd prime and every integer , there is a Heisenberg group of order that has pairwise nonisomorphic quotients of order . Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most . They are also directly and centrally indecomposable and of the same indecomposability type. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.

Derived Lengths of Solvable Groups Having Five Irreducible Character Degrees II

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.

A solvable group whose character degree graph has diameter 3

We show that there is a solvable group G so that the character degree graph of G has diameter 3.

The vanishing-off subgroup

Abstract In this paper, we define the vanishing-off subgroup of a nonabelian group. We study the structure of the quotient of this subgroup and a central series obtained from this subgroup.

Non-divisibility among character degrees

Abstract In this paper, we study groups for which if 1 < a < b are character degrees, then a does not divide b. We say that these groups have the condition no divisibility among degrees (NDAD). We conjecture that the number of character degrees of a group that satisfies NDAD is bounded and we prove this for solvable groups. More precisely, we prove that solvable groups with NDAD have at most four character degrees and have derived length at most 3. We give a group-theoretic characterization of the solvable groups satisfying NDAD with four character degrees. Since the structure of groups with at most three character degrees is known, these results describe the structure of solvable groups with NDAD.

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 p-group Camina pairs

Abstract. Let be a Camina pair. We prove that G must be a p-group for some prime p. We also prove that . Also, we discuss how one might build examples with , although we are not able to prove the existence of such examples.

DERIVED LENGTHS OF SOLVABLE GROUPS SATISFYING THE ONE-PRIME HYPOTHESIS. III

A finite group G is said to satisfy the one-prime hypothesis if the greatest common divisor of any two distinct irreducible character degrees is either 1 or a prime number. The principal result of this paper is that if G is solvable and satisfies the one-prime hypothesis, and if G has a nonabelian nilpotent homomorphic image, then the derived length of G is at most 4.

Generalizing Camina groups and their character tables

Abstract We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are shared by these generalized Camina groups. In particular, we show that if G is a nilpotent, generalized Camina group then its nilpotence class is at most 3. We use the information we collect about generalized Camina groups with nilpotence class 3 to characterize the character tables of these groups.