Search results for "init"

Products of groups and group classes

1994

Letχ be a Schunck class, and let the finite groupG=AB=BC=AC be the product of two nilpotent subgroupsA andB andχ-subgroupC. If for every common prime divisorp of the orders ofA andB the cyclic group of orderp is anχ-group, thenG is anχ-group. This generalizes earlier results of O. Kegel and F. Peterson. Some related results for groups of the formG=AB=AK=BK, whereK is a nilpotent normal subgroup ofG andA andB areχ-groups for some saturated formationχ, are also proved.

On generalised subnormal subgroups of finite groups

2013

Let be a formation of finite groups. A subgroup M of a finite group G is said to be -normal in G if belongs to . A subgroup U of a finite group G is called a K--subnormal subgroup of G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ … ≤ Un = G such that Ui − 1 is either normal or -normal in Ui, for i = 1, 2, …, n. The K--subnormality could be regarded as the natural extension of the subnormality to formation theory and plays an important role in the structural study of finite groups. The main purpose of this paper is to analyse classes of finite groups whose K--subnormal subgroups are exactly the subnormal ones. Some interesting extensions of well-known classes of groups emerge.

About Quotient Orders and Ordering Sequences

2017

Summary In preparation for the formalization in Mizar [4] of lotteries as given in [14], this article closes some gaps in the Mizar Mathematical Library (MML) regarding relational structures. The quotient order is introduced by the equivalence relation identifying two elements x, y of a preorder as equivalent if x ⩽ y and y ⩽ x. This concept is known (see e.g. chapter 5 of [19]) and was first introduced into the MML in [13] and that work is incorporated here. Furthermore given a set A, partition D of A and a finite-support function f : A → ℝ, a function Σ f : D → ℝ, Σ f (X)= ∑ x∈X f(x) can be defined as some kind of natural “restriction” from f to D. The first main result of this article ca…

Some considerations on Hydra groups and a new bound for the length of words

2014

Abstract After a survey on some recent results of Riley and others on Ackermann functions and Hydra groups, we make an analogy between DNA sequences, whose growth is the same of that of Hydra groups, and a musical piece, written with the same algorithmic criterion. This is mainly an aesthetic observation, which emphasizes the importance of the combinatorics of words in two different contexts. A result of specific mathematical interest is placed at the end, where we sharpen some previous bounds on deterministic finite automata in which there are languages with hairpins.

Multi-letter reversible and quantum finite automata

2007

The regular language (a+b)*a (the words in alphabet {a, b} having a as the last letter) is at the moment a classical example of a language not recognizable by a one-way quantum finite automaton (QFA). Up to now, there have been introduced many different models of QFAs, with increasing capabilities, but none of them can cope with this language. We introduce a new, quite simple modification of the QFA model (actually even a deterministic reversible FA model) which is able to recognize this language. We also completely characterise the set of languages recognizable by the new model FAs, by finding a "forbidden construction" whose presence or absence in the minimal deterministic (not necessaril…

Products of pairwise totally permutable groups

2003

[EN] In this paper finite groups factorized as products of pairwise totally permutable subgroups are studied in the framework of Fitting classes

A remark on conjectures in modular representation theory

1987

On a problem of L.A. Shemetkov on superradical formations of finite groups

2014

Abstract Subgroup-closed saturated formations F which are closed under taking products of F -subnormal F -subgroups are studied in the paper. Our results can be regarded as further developments in the hunt for a solution of a problem proposed by L.A. Shemetkov in 1999 in the Kourovka Notebook.

On formations of finite groups with the generalized Wielandt property for residuals II

2018

A formation [Formula: see text] of finite groups has the generalized Wielandt property for residuals, or [Formula: see text] is a GWP-formation, if the [Formula: see text]-residual of a group generated by two [Formula: see text]-subnormal subgroups is the subgroup generated by their [Formula: see text]-residuals. The main result of this paper describes a large family of GWP-formations to further the transparence of this kind of formations, and it can be regarded as a natural step toward the solution of the classification problem.

Degrees of Characters and Values on Prime Order Elements

2008

Two irreducible characters of a finite group with the same value on prime elements have the same degree.