On characters in the principal 2-block, II (original) (raw)

Journal of the Australian Mathematical Society, 1978

Let k be a complex number and let u be an element of a finite group G. Suppose that u does not belong to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(l) -X(u) = fc for every complex nonprincipal irreducible character X in the principal 2-block of G if and only if G/O(G) is isomorphic either to C 2 , a cyclic group of order 2, or to P5L(2,2"),

On character values in finite groups

Bulletin of the Australian Mathematical Society, 1977

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) -X(u) = c or X(u) = 0 . It is shown that c is an even positive integer and all such groups with a -8 are described.

Character values of groups of odd order and a question of Feit

Journal of Algebra, 1981

Let x be an irreducible complex character of a finite group G and let Q&) denote the field obtained by adjoining the values of x to the rational field Q. Let Q, denote the field obtained by adjoining a primitive mth root of unity to Q. We say that x requires mth roots of unity if Qk) is contained in Q, and m is the smallest positive integer with this property. The following open question is raised by Feit 12, p. 411. Suppose that x requires mth roots of unity. Is it true that G contains an element of order m? The purpose of this paper is to provide an affirmative answer to the question in the case that G has odd order.

A new characterization for the simple group PSL(2, p 2) by order and some character degrees

Czechoslovak Mathematical Journal, 2015

Let G be a finite group and p a prime number. We prove that if G is a finite group of order |PSL(2, p 2 )| such that G has an irreducible character of degree p 2 and we know that G has no irreducible character θ such that 2p | θ(1), then G is isomorphic to PSL(2, p 2 ). As a consequence of our result we prove that PSL(2, p 2 ) is uniquely determined by the structure of its complex group algebra.

A Characterisation of Certain Finite Groups of Odd Order

Mathematical Proceedings of the Royal Irish Academy, 2011

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The main object of this paper is to obtain a characterization for all finite groups of odd order with commutativity degree greater than or equal to 11 75 .

Blocks of small defect in alternating groups and squares of Brauer character degrees

Journal of Group Theory, 2017

Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p-subgroup P and G / P {G/P} is nilpotent if and only if φ ⁢ ( 1 ) 2 {\varphi(1)^{2}} divides | G : ker ( φ ) | {|G:{\rm ker}(\varphi)|} for every irreducible Brauer character φ of G.

THE CHARACTERS AND COMMUTATORS OF FINITE GROUPS By TIM W. BONNER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

2009

of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE CHARACTERS AND COMMUTATORS OF FINITE GROUPS By Tim W. Bonner August 2009 Chair: Alexandre Turull Major: Mathematics Let G be a finite group. It is well-known that the elements of the commutator subgroup must be products of commutators, but need not themselves be commutators. A natural question is to determine the minimal integer, λ(G), such that each element of the commutator subgroup may be represented as a product of λ(G) commutators. An analysis of a known character identity allows us to improve the existing lower bounds for |G| in terms of λ(G). The techniques we develop also address the related following question. Suppose we have a conjugacy class C of a finite group G such that 〈C〉 = G = G′. One may ask for the minimal integer cn(C) such that each element of G may be expressed as a product of cn(C) elements of the c...

On Distinct Character Degrees

2022

Berkovich, Chillag and Herzog characterized all finite groups GGG in which all the nonlinear irreducible characters of GGG have distinct degrees. In this paper we extend this result showing that a similar characterization holds for all finite solvable groups GGG that contain a normal subgroup NNN, such that all the irreducible characters of GGG that do not contain NNN in their kernel have distinct degrees.