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1. CMB 2014 (vol 57 pp. 648)
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 |
2. CMB 2014 (vol 57 pp. 277)
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 |
3. CMB 1997 (vol 40 pp. 47)
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 |