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Search: All articles in the CMB digital archive with keyword subgroups

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1. CMB Online first

Akbari, Saieed; Miraftab, Babak; Nikandish, Reza
Co-Maximal Graphs of Subgroups of Groups
Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma(H)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In this paper, we study the connectivity, diameter, clique number and vertex chromatic number of $\Gamma(H)$. For instance, we show that if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$ is connected with diameter at most $3$. Also, we characterize all finite groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma(H)$ is connected and moreover the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.

Keywords:co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable group
Categories:05C25, 05E15, 20D10, 20D15

2. CMB 2014 (vol 57 pp. 648)

Tang, Juping; Miao, Long
On the ${\mathcal F}{\Phi}$-Hypercentre of Finite Groups
Let $G$ be a finite group, $\mathcal F$ a class of groups. Then $Z_{{\mathcal F}{\Phi}}(G)$ is the ${\mathcal F}{\Phi}$-hypercentre of $G$ which is the product of all normal subgroups of $G$ whose non-Frattini $G$-chief factors are $\mathcal F$-central in $G$. A subgroup $H$ is called $\mathcal M$-supplemented in a finite group $G$, if there exists a subgroup $B$ of $G$ such that $G=HB$ and $H_1B$ is a proper subgroup of $G$ for any maximal subgroup $H_1$ of $H$. The main purpose of this paper is to prove: Let $E$ be a normal subgroup of a group $G$. Suppose that every noncyclic Sylow subgroup $P$ of $F^{*}(E)$ has a subgroup $D$ such that $1\lt |D|\lt |P|$ and every subgroup $H$ of $P$ with order $|H|=|D|$ is $\mathcal M$-supplemented in $G$, then $E\leq Z_{{\mathcal U}{\Phi}}(G)$.

Keywords:${\mathcal F}{\Phi}$-hypercentre, Sylow subgroups, $\mathcal M$-supplemented subgroups, formation
Categories:20D10, 20D20

3. CMB 2014 (vol 57 pp. 277)

Elkholy, A. M.; El-Latif, M. H. Abd
On Mutually $m$-permutable Product of Smooth Groups
Let $G$ be a finite group and $H$, $K$ two subgroups of G. A group $G$ is said to be a mutually m-permutable product of $H$ and $K$ if $G=HK$ and every maximal subgroup of $H$ permutes with $K$ and every maximal subgroup of $K$ permutes with $H$. In this paper, we investigate the structure of a finite group which is a mutually m-permutable product of two subgroups under the assumption that its maximal subgroups are totally smooth.

Keywords:permutable subgroups, $m$-permutable, smooth groups, subgroup lattices
Categories:20D10, 20D20, 20E15, 20F16

4. CMB 1997 (vol 40 pp. 47)

Hartl, Manfred
A universal coefficient decomposition for subgroups induced by submodules of group algebras
Dimension subgroups and Lie dimension subgroups are known to satisfy a `universal coefficient decomposition', {\it i.e.} their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the `universal' coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary $N$-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods.

Keywords:induced subgroups, group algebras, Fox subgroups, relative dimension, subgroups, polynomial ideals
Categories:20C07, 16A27

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