Solvable Groups with a Small Number of Prime Divisors in the Element Orders

Keller, Thomas Michael

doi:10.1006/jabr.1994.1357

Cited by 11 publications

(16 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By the previous examples and the main results of [9,10], as well as the main result of [8], Theorem 1.1 can be easily checked. Note that all those upper bounds are the best possible.…”

Section: Proof It Can Be Checked Thatmentioning

confidence: 70%

See 1 more Smart Citation

Two results on character codegrees

Liu,

Yang

2021

Preprint

View full text Add to dashboard Cite

Let G be a finite group and Irr(G) be the set of irreducible characters of G. The codegree of an irreducible character χ of the group G is defined as cod(χ) = |G : ker(χ)|/χ(1). In this paper, we study two topics related to the character codegrees. Let σ c (G) be the maximal integer m such that there is a member in cod(G) having m distinct prime divisors, where cod(G) = {cod(χ)|χ ∈ Irr(G)}. One is related to the codegree version of the Huppert's ρ-σ conjecture and we obtain the best possible bound for |π(G)| under the condition σ c (G) = 2, 3, and 4 respectively. The other 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.

show abstract

“…By the previous examples and the main results of [9,10], as well as the main result of [8], Theorem 1.1 can be easily checked. Note that all those upper bounds are the best possible.…”

Section: Proof It Can Be Checked Thatmentioning

confidence: 70%

“…Let G be a group and N ✂G. For χ ∈ Irr(G), Irr(χ| N ) is the set of irreducible constitutes of the restriction of χ to N. Lemma 2.1 (Lemma 1.8 of [9]).…”

Section: The Codegree Analogue Of the Huppert's ρ-σ Conjecturementioning

confidence: 99%

Two results on character codegrees

Liu,

Yang

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In Section 6, we have Theorem 6.6 and Theorem 6.8 that give a beautiful but uncomplete description of prime graphs of pseudo-solvable groups. It is natural to wonder what the conditions on the prime graphs would become if we allow, instead of A 5 as in the discussion of Section 6, other simple groups, such as PSL (2,7) or Suzuki groups, in the composition series.…”

Section: Discussionmentioning

confidence: 99%

The Prime Graphs of Some Classes of Finite Groups

Florez,

Higgins,

Huang

et al. 2021

Preprint

View full text Add to dashboard Cite

In this paper we study prime graphs of finite groups. The prime graph of a finite group G, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing |G|} and an edge p-q if and only if there exists an element of order pq in G. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any n ∈ N, solvable groups of n th -power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or A5 and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.

show abstract

“…The group that shows (1) appears in the example on page 43 of [15]. The groups that show (2) and (3) appear on p. 632 and p. 631 of [7], respectively. The proof is a tedious analysis of the kernels and degrees of all the irreducible characters of each of these groups.…”

Section: Proofsmentioning

confidence: 99%

Huppert's conjecture for character codegrees

Moretó

2021

Mathematische Nachrichten

View full text Add to dashboard Cite

Huppert's ρ‐σ conjecture is one of the main open problems on character degrees of finite groups. A number of ρ‐σ problems have been studied. For instance, T. Keller and J. Zhang considered the ρ‐σ problem for element orders in the 1990s. Recently, a lot of research is being done on character codegrees. Y. Yang and G. Qian studied the ρ‐σ problem for character codegrees in 2017. In this note, we obtain a sharp bound for groups with trivial solvable radical. As a consequence, we improve the general bound of Yang and Qian. For solvable groups, we notice that the ρ‐σ problem for character codegrees is very closely related to the ρ‐σ problem for element orders. In particular, we give a partial negative answer to a speculation by Yang and Qian on the exact bound for the ρ‐σ problem for character codegrees.

show abstract

Solvable Groups with a Small Number of Prime Divisors in the Element Orders

Cited by 11 publications

References 0 publications

Two results on character codegrees

Two results on character codegrees

The Prime Graphs of Some Classes of Finite Groups

Huppert's conjecture for character codegrees

Contact Info

Product

Resources

About