2018-06-22T16:41:32Z
http://ijgt.ui.ac.ir/?_action=export&rf=summon&issue=4093
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2018
7
2
Sylow multiplicities in finite groups
Dan
Levy
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 $leftvert Grightvert $. The Sylow multiplicity of $gin G$ 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.
Sylow sequences
Sylow multiplicities
Solvable radical
Solvable residual
2018
06
01
1
8
http://ijgt.ui.ac.ir/article_21482_4d16a7d4c6f2488422da19da3ac6bcf6.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2018
7
2
Some characterisations of groups in which normality is a transitive relation by means of subgroup embedding properties
Ramon
Esteban-Romero
Giovanni
Vincenzi
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.
group
subgroup embedding property
T-group
FC$^*$-group
group without infinite simple sections
2018
06
01
9
16
http://ijgt.ui.ac.ir/article_21214_cc340e8f12a5cd3c8681717909e787b8.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2018
7
2
On finite groups with square-free conjugacy class sizes
Maria-Jose
Felipe
Ana
Martinez-Pastor
Victor-Manuel
Ortiz-Sotomayor
We report on finite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.
Finite groups
Conjugacy classes
Factorised groups
2018
06
01
17
24
http://ijgt.ui.ac.ir/article_21475_3c37cb4c86113971ca7d07fb160d560a.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2018
7
2
On metacyclic subgroups of finite groups
Adolfo
Ballester-Bolinches
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.
Finite group
Sylow subgroups
metacyclic groups
2-generator groups
2018
06
01
25
29
http://ijgt.ui.ac.ir/article_21480_b9d4162cb9b5fc9711b0f39d833cf4e0.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2018
7
2
Representations of group rings and groups
Ted
Hurley
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 $Amapsto P^{-1}AP$ (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.
group
ring
Representation
2018
06
01
31
44
http://ijgt.ui.ac.ir/article_21484_2d64c759c091beeacf98923cde8ed7f6.pdf
International Journal of Group Theory
Int. J. Group Theory
2251-7650
2251-7650
2018
7
2
On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras
Nil
Mansuroğlu
Let $L$ be a free Lie algebra of rank $rgeq2$ 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 $2$.
Free Lie algebra
homogeneous and fine homogeneous components
free centre-by-metabelian Lie algebra
second derived ideal
2018
06
01
45
50
http://ijgt.ui.ac.ir/article_21481_0b392ad1ffab7cd79272442ecc91712c.pdf