@Article{Levy2018,
author="Levy, Dan",
title="Sylow multiplicities in finite groups",
journal="International Journal of Group Theory",
year="2018",
volume="7",
number="2",
pages="1-8",
abstract="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.",
issn="2251-7650",
doi="10.22108/ijgt.2017.21482",
url="http://ijgt.ui.ac.ir/article_21482.html"
}
@Article{Esteban-Romero2018,
author="Esteban-Romero, Ramon
and Vincenzi, Giovanni",
title="Some characterisations of groups in which normality is a transitive relation by means of subgroup embedding properties",
journal="International Journal of Group Theory",
year="2018",
volume="7",
number="2",
pages="9-16",
abstract="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.",
issn="2251-7650",
doi="10.22108/ijgt.2017.21214",
url="http://ijgt.ui.ac.ir/article_21214.html"
}
@Article{Felipe2018,
author="Felipe, Maria-Jose
and Martinez-Pastor, Ana
and Ortiz-Sotomayor, Victor-Manuel",
title="On finite groups with square-free conjugacy class sizes",
journal="International Journal of Group Theory",
year="2018",
volume="7",
number="2",
pages="17-24",
abstract="We report on finite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.",
issn="2251-7650",
doi="10.22108/ijgt.2017.21475",
url="http://ijgt.ui.ac.ir/article_21475.html"
}
@Article{Ballester-Bolinches2018,
author="Ballester-Bolinches, Adolfo",
title="On metacyclic subgroups of finite groups",
journal="International Journal of Group Theory",
year="2018",
volume="7",
number="2",
pages="25-29",
abstract="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.",
issn="2251-7650",
doi="10.22108/ijgt.2017.21480",
url="http://ijgt.ui.ac.ir/article_21480.html"
}
@Article{Hurley2018,
author="Hurley, Ted",
title="Representations of group rings and groups",
journal="International Journal of Group Theory",
year="2018",
volume="7",
number="2",
pages="31-44",
abstract="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.",
issn="2251-7650",
doi="10.22108/ijgt.2017.21484",
url="http://ijgt.ui.ac.ir/article_21484.html"
}
@Article{Mansuroğlu2018,
author="Mansuroğlu, Nil",
title="On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras",
journal="International Journal of Group Theory",
year="2018",
volume="7",
number="2",
pages="45-50",
abstract="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 $2$.",
issn="2251-7650",
doi="10.22108/ijgt.2017.21481",
url="http://ijgt.ui.ac.ir/article_21481.html"
}