An Overview of Graphs Associated with Character Degrees And Conjugacy Class Sizes in Finite Groups

Lewis, Mark L.

doi:10.1216/rmj-2008-38-1-175

Cited by 102 publications

(91 citation statements)

References 68 publications

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“…He suggested that it is likely that there is a constant bound and that, in fact, he was not aware of any solvable group G with bounded Fitting height graph and h(G) > 4. (This fact has now been formally stated as a conjecture in [4]. )…”

Section: Introductionmentioning

confidence: 79%

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Fitting Height and Character Degree Graphs

Moretó

2007

Algebr Represent Theor

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Given a group G, (G) is the graph whose vertices are the primes that divide the degree of some irreducible character and two vertices p and q are joined by an edge if pq divides the degree of some irreducible character of G. By a definition of Lewis, a graph has bounded Fitting height if the Fitting height of any solvable group G with (G) = is bounded (in terms of ). In this note, we prove that there exists a universal constant C such that if has bounded Fitting height and (G) = then h(G) ≤ C. This solves a problem raised by Lewis.

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Section: Introductionmentioning

confidence: 79%

“…This graph has been widely studied. For the most recent account of results on this graph see [4]. This graph tends to have many edges.…”

Section: Introductionmentioning

confidence: 99%

Fitting Height and Character Degree Graphs

Moretó

2007

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“…There is an edge between two character degrees a and b if the greatest common divisor of a and b is greater than 1. Many of the results known about these two graphs can be found in the expository paper [8]. While the graphs are different, there are similarities.…”

Section: The Graphsmentioning

confidence: 99%

The Taketa Problem and Character Degree Graphs with Diameter Three

Lewis

Sass²

2015

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“…It is not surprising that these two graphs are closely related. It is not difficult to show that one is connected if and only if the other is; and in this case that their diameters differ by at most 1 (see [8]). Recently, it has been shown that ∆(G) is a complete graph for most simple groups (see [13], [14], and [15]), which perhaps makes it surprising that Γ(G) is never a complete graph for a nonsolvable group G.…”

Section: Main Theorem If γ(G) Is a Complete Graph Then G Is A Solvamentioning

confidence: 99%