Search results for "Order theory"

VARIATIONS ON THOMPSON'S CHARACTER DEGREE THEOREM

2001

If P is a Sylow- p -subgroup of a finite p -solvable group G , we prove that G^\prime \cap \bf{N}_G(P) \subseteq {P} if and only if p divides the degree of every irreducible non-linear p -Brauer character of G. More generally if π is a set of primes containing p and G is π-separable, we give necessary and sufficient group theoretic conditions for the degree of every irreducible non-linear p -Brauer character to be divisible by some prime in π. This can also be applied to degrees of ordinary characters.

HEIGHTS OF CHARACTERS IN BLOCKS OF $p$-SOLVABLE GROUPS

2005

In this paper, it is proved that if $B$ is a Brauer $p$ -block of a $p$ -solvable group, for some odd prime $p$ , then the height of any ordinary character in $B$ is at most $2b$ , where $p^b$ is the largest degree of the irreducible characters of the defect group of $B$ . Some other results that relate the heights of characters with properties of the defect group are obtained.

A local approach to a class of locally finite groups

2003

This paper is devoted to the study of a class of generalised P-nilpotent groups in the universe cℒ̄ of all radical locally finite groups satisfying min-q for every prime q. Some results of finite groups are extended and a characterisation of the injectors associated with this class is given.

Correspondences Between 2-Brauer Characters of Solvable Groups

2010

Let G be a finite solvable group and let p be a prime. Let P ∈ Syl p (G) and N = N G (P). We prove that there exists a natural bijection between the 2-Brauer irreducible characters of p′-degree of G and those of N G (P).

On the number of prime divisors of the order of elliptic curves modulo p

2005

BOUNDING THE NUMBER OF IRREDUCIBLE CHARACTER DEGREES OF A FINITE GROUP IN TERMS OF THE LARGEST DEGREE

2013

We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.

Ambainis-Freivalds’ Algorithm for Measure-Once Automata

2001

An algorithm given by Ambainis and Freivalds [1] constructs a quantum finite automaton (QFA) with O(log p) states recognizing the language Lp = {ai| i is divisible by p} with probability 1 - Ɛ , for any Ɛ > 0 and arbitrary prime p. In [4] we gave examples showing that the algorithm is applicable also to quantum automata of very limited size. However, the Ambainis-Freivalds algoritm is tailored to constructing a measure-many QFA (defined by Kondacs andWatrous [2]), which cannot be implemented on existing quantum computers. In this paper we modify the algorithm to construct a measure-once QFA of Moore and Crutchfield [3] and give examples of parameters for this automaton. We show for the lang…

Elementarteiler von Inzidenzmatrizen symmetrischer Blockpläne

1986

By a study of the integral code generated by the rows of the incidence matrix and its extention the following results are obtained: Let d 1,...,d V(d 1|d 2,d 2|d 3...) be the elementary divisors of the incidence matrix of a symmetric (v,n+λ, λ) design. Then d v=(n+λ)n/g.c.d. (n, λ). Moreover, if p is a prime such that p|n, p∤λ and if x p denotes the p-part of x, then (d idv+2−i) p =n p for 2≤i≤v. For projective planes it can be shown that d 1=···=d 3n−2=1, hence $$d_{n^2 - 2n{\text{ }} + {\text{ }}5} {\text{ }} = \cdots = d_{n^2 + n} = n$$ and $$d_{n^2 - n{\text{ }} + {\text{ }}1} = (n + 1)n$$ . The paper also contains some results about elementary divisors of incidence matrices G satisfyin…

Periodic Solutions of the Second Order Quadratic Rational Difference Equation $$x_{n+1}=\frac{\alpha }{(1+x_n)x_{n-1}} $$ x n + 1 = α ( 1 + x n ) x n…

2016

The aim of this article is to investigate the periodic nature of solutions of a rational difference equation $$x_{n+1}=\frac{\alpha }{(1+x_n)x_{n-1}}. {(*)} $$ We explore Open Problem 3.3 given in Amleh et al. (Int J Differ Equ 3(1):1–35, 2008, [2]) that requires to determine all periodic solutions of the equation (*). We conclude that for the equation (*) there are no periodic solution with prime period 3 and 4. Period 7 is first period for which exists nonnegative parameter \(\alpha \) and nonnegative initial conditions.

p-Parts of Brauer character degrees

2014

Abstract Let G be a finite group and let p be an odd prime. Under certain conditions on the p-parts of the degrees of its irreducible p-Brauer characters, we prove the solvability of G. As a consequence, we answer a question proposed by B. Huppert in 1991: If G has exactly two distinct irreducible p-Brauer character degrees, then is G solvable? We also determine the structure of non-solvable groups with exactly two irreducible 2-Brauer character degrees.