Group rings for communications
Ted
Hurley
National University of Ireland Galway
author
text
article
2015
eng
This is a survey of some recent applications of abstract algebra, and in particular group rings, to the communications' areas.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
4
no.
2015
1
23
http://ijgt.ui.ac.ir/article_5453_9f9342e7224465dd42f3537a6a7fe39a.pdf
dx.doi.org/10.22108/ijgt.2015.5453
Groups of order $p^8$ and exponent $p$
Michael
Vaughan-Lee
Oxford University
Mathematical Institute
author
text
article
2015
eng
We prove that for $p>7$ there are \[ p^{4}+2p^{3}+20p^{2}+147p+(3p+29)\gcd (p-1,3)+5\gcd (p-1,4)+1246 \] groups of order $p^{8}$ with exponent $p$. If $P$ is a group of order $p^{8}$ and exponent $p$, and if $P$ has class $c>1$ then $P$ is a descendant of $ P/\gamma _{c}(P)$. For each group of exponent $p$ with order less than $ p^{8} $ we calculate the number of descendants of order $p^{8}$ with exponent $p$. In all but one case we are able to obtain a complete and irredundant list of the descendants. But in the case of the three generator class two group of order $p^{6}$ and exponent $p$ ($p>3$), while we are able to calculate the number of descendants of order $p^{8}$, we have not been able to obtain a list of the descendants.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
4
no.
2015
25
42
http://ijgt.ui.ac.ir/article_5758_388335a7f7fa07bfe83e3a007db2f69c.pdf
dx.doi.org/10.22108/ijgt.2015.5758
Modules over group rings of groups with restrictions on the system of all proper subgroups
Olga
Dashkova
Professor of the Branch of Moscow state university in Sevastopol
author
text
article
2015
eng
We consider the class $\mathfrak M$ of $\bf R$--modules where $\bf R$ is an associative ring. Let $A$ be a module over a group ring $\bf R$$G$, $G$ be a group and let $\mathfrak L(G)$ be the set of all proper subgroups of $G$. We suppose that if $H \in \mathfrak L(G)$ then $A/C_{A}(H)$ belongs to $\mathfrak M$. We investigate an $\bf R$$G$--module $A$ such that $G \not = G'$, $C_{G}(A) = 1$. We study the cases: 1) $\mathfrak M$ is the class of all artinian $\bf R$--modules, $\bf R$ is either the ring of integers or the ring of $p$--adic integers; 2) $\mathfrak M$ is the class of all finite $\bf R$--modules, $\bf R$ is an associative ring; 3) $\mathfrak M$ is the class of all finite $\bf R$--modules, $\bf R$$=F$ is a finite field.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
4
no.
2015
43
48
http://ijgt.ui.ac.ir/article_5504_b2fda9c27d76f5bae4d99c57c8252f39.pdf
dx.doi.org/10.22108/ijgt.2015.5504
Sylow like theorems for $V(\mathbb{Z}G)$
Wolfgang
Kimmerle
University of Stuttgart
author
text
article
2015
eng
The main part of this article is a survey on torsion subgroups of the unit group of an integral group ring. It contains the major parts of my talk given at the conference "Groups, Group Rings and Related Topics" at UAEU in Al Ain October 2013. In the second part special emphasis is layed on $p$ - subgroups and on the open question whether there is a Sylow like theorem in the normalized unit group of an integral group ring. For specific classes of finite groups we prove that $p$ - subgroups of the normalized unit group of its integral group rings $V(\mathbb{Z}G)$ are isomorphic to subgroups of $G .$ In particular for $p = 2$ this is shown when $G$ has abelian Sylow $2$ - subgroups. This extends results known for soluble groups to classes of groups which are not contained in the class of soluble groups.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
4
no.
2015
49
59
http://ijgt.ui.ac.ir/article_5452_1c109992064605dc78186ce51e795fb6.pdf
dx.doi.org/10.22108/ijgt.2015.5452
On group rings and some of their applications to combinatorics and symmetric cryptography
Claude
Carlet
Universities of Paris 8 and Paris 13
author
Yin
Tan
University of Waterloo
author
text
article
2015
eng
We give a survey of recent applications of group rings to combinatorics and to cryptography, including their use in the differential cryptanalysis of block ciphers.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
4
no.
2015
61
74
http://ijgt.ui.ac.ir/article_5813_aae52578d139936ffc37b6d71a0be349.pdf
dx.doi.org/10.22108/ijgt.2015.5813