Sylow multiplicities in finite groups
Dan
Levy
Italy
author
text
article
2018
eng
Let $G$ be a finite group and let $\mathcal{P}=P_{1},\ldots,P_{m}$ be a sequence of Sylow $p_{i}$-subgroups of $G$, where $p_{1},\ldots,p_{m}$ are the distinct prime divisors of $\left\vert G\right\vert $. The Sylow multiplicity of $g\in G$ in $\mathcal{P}$ is the number of distinct factorizations $g=g_{1}\cdots g_{m}$ such that $g_{i}\in P_{i}$. We review properties of the solvable radical and the solvable residual of $G$ which are formulated in terms of Sylow multiplicities, and discuss some related open questions.
International Journal of Group Theory
University of Isfahan
2251-7650
7
v.
2
no.
2018
1
8
http://ijgt.ui.ac.ir/article_21482_4d16a7d4c6f2488422da19da3ac6bcf6.pdf
dx.doi.org/10.22108/ijgt.2017.21482
Some characterisations of groups in which normality is a transitive relation by means of subgroup embedding properties
Ramon
Esteban-Romero
Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Camí de Vera, s/n, 46022 València, Spain
author
Giovanni
Vincenzi
University of salerno
author
text
article
2018
eng
In this survey we highlight the relations between some subgroup embedding properties that characterise groups in which normality is a transitive relation in certain universes of groups with some finiteness properties.
International Journal of Group Theory
University of Isfahan
2251-7650
7
v.
2
no.
2018
9
16
http://ijgt.ui.ac.ir/article_21214_cc340e8f12a5cd3c8681717909e787b8.pdf
dx.doi.org/10.22108/ijgt.2017.21214
On finite groups with square-free conjugacy class sizes
Maria-Jose
Felipe
Universidad Politecnica de Valencia
author
Ana
Martinez-Pastor
Universidad Politecnica de Valencia
author
Victor-Manuel
Ortiz-Sotomayor
Universidad Politecnica de Valencia
author
text
article
2018
eng
We report on finite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.
International Journal of Group Theory
University of Isfahan
2251-7650
7
v.
2
no.
2018
17
24
http://ijgt.ui.ac.ir/article_21475_3c37cb4c86113971ca7d07fb160d560a.pdf
dx.doi.org/10.22108/ijgt.2017.21475
On metacyclic subgroups of finite groups
Adolfo
Ballester-Bolinches
Departament de Matematiques, Universitat de Valencia, Burjassot, Valencia, Spain
author
text
article
2018
eng
The aim of this survey article is to present some structural results about of groups whose Sylow p-subgroups are metacylic (p a prime). A complete characterisation of non-nilpotent groups whose 2-generator subgroups are metacyclic is also presented.
International Journal of Group Theory
University of Isfahan
2251-7650
7
v.
2
no.
2018
25
29
http://ijgt.ui.ac.ir/article_21480_b9d4162cb9b5fc9711b0f39d833cf4e0.pdf
dx.doi.org/10.22108/ijgt.2017.21480
Representations of group rings and groups
Ted
Hurley
National University of Ireland Galway
author
text
article
2018
eng
An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. It is shown that for any group ring matrix $A$ of $\mathbb{C} G$ there exists a matrix $U$ (independent of $A$) such that $U^{-1}AU= diag(T_1,T_2,\ldots, T_r)$ for block matrices $T_i$ of fixed size $s_i × s_i$ where $r$ is the number of conjugacy classes of $G$ and $s_i$ are the ranks of the group ring matrices of the primitive idempotents. Using the isomorphism of the group ring to the ring of group ring matrices followed by the mapping $A\mapsto P^{-1}AP$ (fixed $P$) gives an isomorphism from the group ring to the ring of such block matrices. Specialising to the group elements gives a faithful representation of the group. Other representations of $G$ may be derived using the blocks in the images of the group elements. For a finite abelian group $Q$ an explicit matrix $P$ is given which diagonalises any group ring matrix of $\mathbb{C}Q$. The characters of $Q$ and the character table of $Q$ may be read off directly from the rows of the diagonalising matrix $P$. This is a special case of the general block diagonalisation process but is arrived at independently. The case for cyclic groups is well-known: Circulant matrices are the group ring matrices of the cyclic group and the Fourier matrix diagonalises any circulant matrix. This has applications to signal processing.
International Journal of Group Theory
University of Isfahan
2251-7650
7
v.
2
no.
2018
31
44
http://ijgt.ui.ac.ir/article_21484_2d64c759c091beeacf98923cde8ed7f6.pdf
dx.doi.org/10.22108/ijgt.2017.21484
On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras
Nil
Mansuroğlu
Ahi Evran University
author
text
article
2018
eng
Let $L$ be a free Lie algebra of rank $r\geq2$ over a field $F$ and let $L_n$ denote the degree $n$ homogeneous component of $L$. By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field $F$, we determine the dimension of $[L_2,L_2,L_1]$. Moreover, by this method, we show that the dimension of $[L_2,L_2,L_1]$ over a field of characteristic $2$ is different from the dimension over a field of characteristic other than $2$.
International Journal of Group Theory
University of Isfahan
2251-7650
7
v.
2
no.
2018
45
50
http://ijgt.ui.ac.ir/article_21481_0b392ad1ffab7cd79272442ecc91712c.pdf
dx.doi.org/10.22108/ijgt.2017.21481