scholarly journals Finite groups with a faithful real-valued irreducible character whose square has exactly two distinct irreducible constituents

2010 ◽  
Vol 45 (1) ◽  
pp. 85-92
Author(s):  
Emmanuel Zhmud
Keyword(s):  
Finite Groups ◽  
Irreducible Character

A NOTE ON THE NUMBER OF ZEROS IN THE COLUMNS OF THE CHARACTER TABLE OF A GROUP

2012 ◽  
Vol 12 (02) ◽  
pp. 1250150 ◽  
Author(s):  
JINSHAN ZHANG ◽  
ZHENCAI SHEN ◽  
SHULIN WU
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Conjugacy Classes ◽  
Character Table ◽  
Number Of Zeros

The finite groups in which every irreducible character vanishes on at most three conjugacy classes were characterized [J. Group Theory13 (2010) 799–819]. Dually, we investigate the finite groups whose columns contain a small number of zeros in the character table.

On finite groups with large degrees of irreducible character

2016 ◽  
Vol 50 (7) ◽  
pp. 497-509 ◽  
Author(s):  
L. S. Kazarin ◽  
S. S. Poiseeva
Keyword(s):  
Finite Groups ◽  
Irreducible Character

Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2

1991 ◽  
Vol 43 (4) ◽  
pp. 792-813 ◽  
Author(s):  
G. O. Michler ◽  
J. B. Olsson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
A Finite Group ◽  
Covering Groups ◽  
Fundamental Paper

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

Coprime Group Actions Fixing All Nonlinear Irreducible Characters

1989 ◽  
Vol 41 (1) ◽  
pp. 68-82 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Group Actions ◽  
Character Degrees ◽  
Prime Divisors ◽  
Irreducible Characters ◽  
Nilpotent Length ◽  
Derived Subgroup ◽  
Nonlinear Irreducible Character

The main result of this paper is the following:Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the setcd(G) = ﹛x(l)lx ∈ Irr(G) ﹜of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.

Orbits and Stabilizers for Solvable Linear Groups

2006 ◽  
Vol 49 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Irreducible Character ◽  
General Class ◽  
Frobenius Group ◽  
Solvable Groups ◽  
Character Degrees ◽  
Linear Groups ◽  
Finite Vector ◽  
Finite Vector Space

AbstractWe extend a result of Noritzsch, which describes the orbit sizes in the action of a Frobenius group G on a finite vector space V under certain conditions, to a more general class of finite solvable groups G. This result has applications in computing irreducible character degrees of finite groups. Another application, proved here, is a result concerning the structure of certain groups with few complex irreducible character degrees.

scholarly journals Word problems for finite nilpotent groups

2020 ◽  
Vol 115 (6) ◽  
pp. 599-609
Author(s):  
Rachel D. Camina ◽  
Ainhoa Iñiguez ◽  
Anitha Thillaisundaram
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Character Degrees ◽  
Character Theory ◽  
Odd Order ◽  
Related Group

AbstractLet w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that $$N_w(1)\ge |G|^{k-1}$$ N w ( 1 ) ≥ | G | k - 1 , where for $$g\in G$$ g ∈ G , the quantity $$N_w(g)$$ N w ( g ) is the number of k-tuples $$(g_1,\ldots ,g_k)\in G^{(k)}$$ ( g 1 , … , g k ) ∈ G ( k ) such that $$w(g_1,\ldots ,g_k)={g}$$ w ( g 1 , … , g k ) = g . Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that $$N_w(g)\ge |G|^{k-1}$$ N w ( g ) ≥ | G | k - 1 for g a w-value in G, and prove that $$N_w(g)\ge |G|^{k-2}$$ N w ( g ) ≥ | G | k - 2 for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

scholarly journals Blocks and Normal Subgroups of Finite Groups

1963 ◽  
Vol 22 ◽  
pp. 15-32 ◽  
Author(s):  
W. F. Reynolds
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Irreducible Character ◽  
Normal Subgroups ◽  
Irreducible Characters ◽  
Section 8 ◽  
A Finite Group

Let H be a normal subgroup of a finite group G, and let ζ be an (absolutely) irreducible character of H. In [7], Clifford studied the irreducible characters X of G whose restrictions to H contain ζ as a constituent. First he reduced this question to the same question in the so-called inertial subgroup S of ζ in G, and secondly he described the situation in S in terms of certain projective characters of S/H. In section 8 of [10], Mackey generalized these results to the situation where all the characters concerned are projective.

scholarly journals Finite groups with an irreducible character of large degree

2015 ◽  
Vol 149 (3-4) ◽  
pp. 523-546 ◽  
Author(s):  
Nguyen Ngoc Hung ◽  
Mark L. Lewis ◽  
Amanda A. Schaeffer Fry
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Large Degree

Finite groups with at most six vanishing conjugacy classes

2021 ◽  
pp. 2250076
Author(s):  
Sajjad M. Robati ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Conjugacy Classes ◽  
Character Formula ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
A Finite Group

Let [Formula: see text] be a finite group. We say that a conjugacy class of [Formula: see text] in [Formula: see text] is vanishing if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

Sign in / Sign up

Export Citation Format

Share Document