Products of primes in conjugacy class sizes and irreducible character degrees

Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows: (1) If gcd (p-1, |G|) = 1 and p2 does not divide |xG| for any p′-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p-subgroup of G/Op(G) is elementary abelian. (2) Suppose that G is p-solvable. If pp-1 does not divide |xG| for any element x of prime power order, then the p-length of G is at most one. (3) Suppose that G is p-solvable. If pp-1 does not divide χ(1) for any χ ∈ Irr (G), then both the p-length and p′-length of G are at most 2.

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Bounding the order of a group with a large conjugacy class

Journal of Group Theory ◽

10.1515/jgth-2014-0033 ◽

2015 ◽

Vol 18 (2) ◽

Author(s):

Anthony W. Harrison

Keyword(s):

Conjugacy Class ◽

Irreducible Character ◽

Character Degrees

AbstractIn this paper, we present results for a group theoretic analog to a parameter defined in terms of irreducible character degrees. Let

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Prime Factors of π-Partial Character Degrees and Conjugacy Class Sizes of π-Elements

Algebra Colloquium ◽

10.1142/s1005386705000660 ◽

2005 ◽

Vol 12 (04) ◽

pp. 699-707

Author(s):

Antonio Beltrán ◽

María José Felipe

Keyword(s):

Conjugacy Class ◽

Solvable Group ◽

Character Degree ◽

Character Degrees ◽

Finite Solvable Group ◽

Conjugacy Class Sizes ◽

Prime Factors

Let G be a finite solvable group. We prove that any prime dividing any irreducible π-partial character degree of G divides the size of some conjugacy class of π-elements of G. Under certain hypothesis, we show that if two distinct primes r and s both divide some irreducible π-partial character degree, then there exists a conjugacy class of π-elements whose size is divisible by rs.

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On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-058-8 ◽

2015 ◽

Vol 58 (1) ◽

pp. 105-109 ◽

Cited By ~ 3

Author(s):

Samaneh Hossein-Zadeh ◽

Ali Iranmanesh ◽

Mohammad Ali Hosseinzadeh ◽

Mark L. Lewis

Keyword(s):

Conjugacy Class ◽

Direct Product ◽

Finite Groups ◽

Character Degrees ◽

Direct Products ◽

Conjugacy Class Sizes ◽

Vertex Graph ◽

The Common ◽

Positive Integers

Abstract.The prime vertex graph, Δ(X), and the common divisor graph, Γ(X), are two graphs that have been deûned on a set of positive integers X. Some properties of these graphs have been studied in the cases where either X is the set of character degrees of a group or X is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.

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