Search results for "Grups"
showing 10 items of 56 documents
Transitivity of Sylow permutability, the converse of Lagrange's theorem, and mutually permutable products
2008
This paper is devoted to the study of mutually permutable products of finite groups. A factorised group G = AB is said to be a mutually permutable product of its factors A and B when each factor permutes with every subgroup of the other factor. We prove that mutually permutable products of Y -groups (groups satisfying the converse of Lagrange's theorem) and SC-groups (groups whose chief factors are simple) are SC -groups. Next, we show that a product of pairwise mutually permutable Y -groups is supersoluble. Finally, we give a local version of the result stating that if a mutually permutable product of two groups is a PST - group (that is, a group in which every subnormal subgroup permutes …
Some subgroup embeddings in finite groups
2015
In this survey paper several subgroup embedding properties related to some types of permutability are introduced and studied.
Real elements and p-nilpotence of finite groups
2016
Our first main result proves that every element of order 4 of a Sylow 2-subgroup S of a minimal non-2-nilpotent group G, is a real element of S. This allows to give a character-free proof of a theorem due to Isaacs and Navarro (see [9, Theorem B]). As an application, the authors show a common extension of the p-nilpotence criteria proved in [3] and [9]. The first and the second authors have been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economía y Competitividad, Spain, and FEDER, European Union. The first author has been also supported by a project from the National Natural Science Foundation of China (NSFC, No. 11271085) and a project of Natural Science Foundation…
Permutable subnormal subgroups of finite groups
2009
The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called PT -groups. In particular, it is shown that the finite solvable PT -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugatepermutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalentl…
On the p-length of some finite p-soluble groups
2014
The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose $p$-length is greater than $1$, $p$ a prime number. Alternative proofs and improvements of recent results about the influence of minimal $p$-subgroups on the $p$-nilpotence and $p$-length of a finite group arise as consequences of our study
p-grupos finitos
1997
In the first part, new bounds for the number of conjugacy classes of maximal length of a finite $p$-group $G$, $r(G)$, are obtained, and they are related with the length of these classes. If $r(G)=p^m-b-1$, there exists a unique normal subgroup of order $p^b$, $N_b$, which is characteristic, and structural properties of these groups are studied when $b\le 3$, by paying special attention to the relation between $N_b$, $Z(G)$ and $G$. In the second part, new bounds for the degree of commutativity of a $p$-group of maximal class are established. In Chapter 2, the bounds known for $c$ are reviewed. In Chapter 3, the reuslts obtained by Shepherd for $c_0\le 4$ are extended to $c_0\le 10$ by mean…
Z-permutable subgroups of finite groups
2016
Let Z be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called Z-permutable if H permutes with all members of Z. The main goal of this paper is to study the embedding of the Z-permutable subgroups and the influence of Z-permutability on the group structure.
Some local properties defining $T_0$-groups and related classes of groups
2016
[EN] We call G a Hall_X -group if there exists a normal nilpotent subgroup N of G for which G/N' is an X -group. We call G a T0 -group provided G/\Phi(G) is a T -group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define Hall_X -groups and T_0 -groups where X ∈ {T , PT , PST }; the classes PT and PST denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.
Sylow permutable subnormal subgroups of finite groups II
2001
[EN] In this paper a local version of Agrawal's theorem about the structure of finite groups in which Sylow permutability is transitive is given. The result is used to obtain new characterisations of this class of finite groups.
Las matemáticas del cubo de Rubik
2013
[ES] En este artículo mostramos cómo podemos utilizar el cubo de Rubik para presentar algunos conceptos básicos de la teoría de grupos y cómo podemos usar esta para resolver el cubo de Rubik.