scholarly journals On the number of conjugacy class sizes and character degrees in finite $p$-groups

2001 ◽  
Vol 129 (11) ◽  
pp. 3201-3204 ◽  
Author(s):  
Gustavo A. Fernández-Alcober ◽  
Alexander Moretó
Keyword(s):  
Conjugacy Class ◽  
Character Degrees ◽  
Conjugacy Class Sizes

ON CONJUGACY CLASS SIZES AND CHARACTER DEGREES OF FINITE GROUPS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350100 ◽  
Author(s):  
GUOHUA QIAN ◽  
YANMING WANG
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Prime Power ◽  
P Element ◽  
Character Degrees ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
A Finite Group

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.

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

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.

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

2015 ◽  
Vol 58 (1) ◽  
pp. 105-109 ◽  
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.

scholarly journals Fixed point spaces, primitive character degrees and conjugacy class sizes

2006 ◽  
Vol 134 (11) ◽  
pp. 3123-3130 ◽  
Author(s):  
I. M. Isaacs ◽  
Thomas Michael Keller ◽  
U. Meierfrankenfeld ◽  
Alexander Moretó
Keyword(s):  
Fixed Point ◽  
Conjugacy Class ◽  
Character Degrees ◽  
Primitive Character ◽  
Conjugacy Class Sizes
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