Finite groups whose minimal subgroups are weakly $\mathcal{H}^{\ast}$-subgroups
Abdelrahman
Heliel
Department of Mathematics, Faculty of Science, Beni-Suef university
author
Rola
Hijazi
Department of Mathematics, Faculty of Science, KAU, Saudi Arabia
author
Reem
Al-Obidy
Department of Mathematics, Faculty of Science, KAU, Saudi Arabia
author
text
article
2014
eng
Let $G$ be a finite group. A subgroup $H$ of $G$ is called an $\mathcal{H}$-subgroup in $G$ if $N_G(H)\cap H^{g}\leq H$ for all $g\in G$. A subgroup $H$ of $G$ is called a weakly $\mathcal{H}^{\ast}$-subgroup in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is an $\mathcal{H}$-subgroup in $G$. We investigate the structure of the finite group $G$ under the assumption that every cyclic subgroup of $G$ of prime order $p$ or of order $4$ (if $p=2$) is a weakly $\mathcal{H}^{\ast}$-subgroup in $G$. Our results improve and extend a series of recent results in the literature.
International Journal of Group Theory
University of Isfahan
2251-7650
3
v.
3
no.
2014
1
11
http://ijgt.ui.ac.ir/article_3837_4ba7139afccee4a6543ffa5a60f76f6d.pdf
dx.doi.org/10.22108/ijgt.2014.3837
The coprime graph of a group
Xuan Long
Ma
Beijing Normal University
author
Hua Quan
Wei
Guangxi University
author
Li Ying
Yang
Guangxi Teachers Education University
author
text
article
2014
eng
The coprime graph $\gg$ with a finite group $G$ as follows: Take $G$ as the vertex set of $\gg$ and join two distinct vertices $u$ and $v$ if $(|u|,|v|)=1$. In the paper, we explore how the graph theoretical properties of $\gg$ can effect on the group theoretical properties of $G$.
International Journal of Group Theory
University of Isfahan
2251-7650
3
v.
3
no.
2014
13
23
http://ijgt.ui.ac.ir/article_4363_c9f7b91082201904334198ebeaf4569a.pdf
dx.doi.org/10.22108/ijgt.2014.4363
Units in $F_{2^k}D_{2n}$
Neha
Makhijani
Indian Institute of Technology Delhi
Hauz Khas, New Delhi-110016
India
author
R.
Sharma
Indian Institute of Technology Delhi
Hauz Khas, New Delhi
India
author
J. B.
Srivastava
Indian Institute of Technology Delhi
Hauz Khas, New Delhi
India
author
text
article
2014
eng
Let $\mathbb{F}_{q}D_{2n}$ be the group algebra of $D_{2n}$, the dihedral group of order $2n$, over $\mathbb{F}_{q}=GF(q)$. In this paper, we establish the structure of $\mathcal{U}(\mathbb{F}_{2^{k}}D_{2n})$, the unit group of $\mathbb{F}_{2^{k}}D_{2n}$ and that of its normalized unitary subgroup $V_{*}(\mathbb{F}_{2^{k}}D_{2n})$ with respect to canonical involution $*$ when $n$ is odd.
International Journal of Group Theory
University of Isfahan
2251-7650
3
v.
3
no.
2014
25
34
http://ijgt.ui.ac.ir/article_4382_381809516a5a923153e331a4094a2b06.pdf
dx.doi.org/10.22108/ijgt.2014.4382
On $n$-Kappe groups
Asadollah
Faramarzi Salles
Damghan University
author
Hassan
Khosravi
Gonbad-e Qabus University
author
text
article
2014
eng
Let $G$ be an infinite group and $n\in \{3, 6\}\cup\{2^k| k\in \mathbb{N}\}$. In this paper, we prove that $G$ is an $n$-Kappe group if and only if for any two infinite subsets $X$ and $Y$ of $G$, there exist $x\in X$ and $y\in Y$ such that $[x^n, y, y]=1$.
International Journal of Group Theory
University of Isfahan
2251-7650
3
v.
3
no.
2014
35
38
http://ijgt.ui.ac.ir/article_4434_9a17cff5f39ae447a1760a128a838980.pdf
dx.doi.org/10.22108/ijgt.2014.4434
Splitting of extensions in the category of locally compact abelian groups
Hossein
Sahleh
Department of Mathematics
University of Guilan
author
Akbar
Alijani
University of Guilan
author
text
article
2014
eng
Let $\pounds$ be the category of all locally compact abelian (LCA) groups. In this paper, the groups $G$ in $\pounds$ are determined such that every extension $0\to X\to Y\to G\to 0$ with divisible, $\sigma-$compact $X$ in $\pounds$ splits. We also determine the discrete or compactly generated LCA groups $H$ such that every pure extension $0\to H\to Y\to X\to 0$ splits for each divisible group $X$ in $\pounds$.
International Journal of Group Theory
University of Isfahan
2251-7650
3
v.
3
no.
2014
39
45
http://ijgt.ui.ac.ir/article_4435_22e1c3d0e09071cafadb580f175adec2.pdf
dx.doi.org/10.22108/ijgt.2014.4435
On the total character of finite groups
Sunil
Prajapati
NBHM Postdoctoral fellow in Indian Statistical Institute Bangalore (I have submitted my PhD thesis at Indian Institute of Technology Delhi).
author
Balasubramanian
Sury
Indian Statistical Institute bangalore, India
author
text
article
2014
eng
For a finite group $G$, we study the total character $\tau_G$ afforded by the direct sum of all the non-isomorphic irreducible complex representations of $G$. We resolve for several classes of groups (the Camina $p$-groups, the generalized Camina $p$-groups, the groups which admit $(G,Z(G))$ as a generalized Camina pair), the problem of existence of a polynomial $f(x) \in \mathbb{Q}[x]$ such that $f(\chi) = \tau_G$ for some irreducible character $\chi$ of $G$. As a consequence, we completely determine the $p$-groups of order at most $p^5$ (with $p$ odd) which admit such a polynomial. We deduce the characterization that these are the groups $G$ for which $Z(G)$ is cyclic and $(G,Z(G))$ is a generalized Camina pair and, we conjecture that this holds good for $p$-groups of any order.
International Journal of Group Theory
University of Isfahan
2251-7650
3
v.
3
no.
2014
47
67
http://ijgt.ui.ac.ir/article_4446_e0d321ff268fc949ff98a187267f48e3.pdf
dx.doi.org/10.22108/ijgt.2014.4446