Search results for "prime"

Prime Factors of Character Degrees of Solvable Groups

1987

Extremal Frobenius numbers in a class of sets

1998

For given $ A_k=\{ a_1,\ldots ,a_k \}, a_1 \le \ldots \le a_k $ coprime the Frobenius number $ {g}(A_k) $ is defined as the greatest integer ${g}$ with no representation¶¶ ${g}=\sum \limits ^k_{i=1}\,x_i\,a_i,\;x_i\in {\Bbb N}_0 $ . ¶¶A class $ {\bf A}^*_k $ is given, such that ¶¶ $ {\overline {g}}^*(k,y):= \max \{ {g}(A_k)|A_k\in {\bf A}^*_k,\, a_k\le y \} $ ¶¶has the same asymptotic behaviour as the general function¶¶ $ {\overline {g}}(k,y):= \max \{ {g}(A_k)| a_k\le y \}\, {\rm for} \, y\to \infty $ .¶¶ Furthermore, ¶¶ $ {\underline {g}}^*(k,x):= \min \{ {g}(A_k)|A_k\in {\bf A}^*_k,\, a_1\ge x \} $ ¶¶is shown to have the same order of magnitude as the general function¶¶ $ {\underline {g}…

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.

On finite products of groups and supersolubility

2010

Two subgroups X and Y of a group G are said to be conditionally permutable in G if X permutes with Y(g) for some element g E G. i.e., XY(g) is a subgroup of G. Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.

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

Saturated formations and products of connected subgroups

2011

Abstract For a non-empty class of groups C , two subgroups A and B of a group G are said to be C -connected if 〈 a , b 〉 ∈ C for all a ∈ A and b ∈ B . Given two sets π and ρ of primes, S π S ρ denotes the class of all finite soluble groups that are extensions of a normal π-subgroup by a ρ-group. It is shown that in a finite group G = A B , with A and B soluble subgroups, then A and B are S π S ρ -connected if and only if O ρ ( B ) centralizes A O π ( G ) / O π ( G ) , O ρ ( A ) centralizes B O π ( G ) / O π ( G ) and G ∈ S π ∪ ρ . Moreover, if in this situation A and B are in S π S ρ , then G is in S π S ρ . This result is then extended to a large family of saturated formations F , the so-c…

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…

Injectors and Radicals in Products of Totally Permutable Groups

2003

Abstract Two subgroups H and K of a group G are said to be totally permutable if every subgroup of H permutes with every subgroup of K. In this paper the behaviour of radicals and injectors associated to Fitting classes in a product of pairwise totally permutable finite groups is studied.