4,593 research outputs found
On the converse of Hall's theorem
In this paper, we mainly investigate the converse of a well-known theorem
proved by P. Hall, and present detailed characterizations under the various
assumptions of the existence of some families of Hall subgroups. In particular,
we prove that if and a finite group has a Hall -subgroup
for every prime , then is -soluble
Electronic Geometry Textbook: A Geometric Textbook Knowledge Management System
Electronic Geometry Textbook is a knowledge management system that manages
geometric textbook knowledge to enable users to construct and share dynamic
geometry textbooks interactively and efficiently. Based on a knowledge base
organizing and storing the knowledge represented in specific languages, the
system implements interfaces for maintaining the data representing that
knowledge as well as relations among those data, for automatically generating
readable documents for viewing or printing, and for automatically discovering
the relations among knowledge data. An interface has been developed for users
to create geometry textbooks with automatic checking, in real time, of the
consistency of the structure of each resulting textbook. By integrating an
external geometric theorem prover and an external dynamic geometry software
package, the system offers the facilities for automatically proving theorems
and generating dynamic figures in the created textbooks. This paper provides a
comprehensive account of the current version of Electronic Geometry Textbook.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
On a problem from the Kourovka Notebook
In this manuscript, a solution to Problem 18.91(b) in the Kourovka Notebook
is given by proving the following theorem. Let be a Sylow -subgroup of a
group with . Suppose that there is an integer such that and every subgroup of of order is -propermutable in ,
and also, in the case that , and is non-abelian, every cyclic
subgroup of of order is -propermutable in . Then is
-nilpotent
On weakly S-embedded subgroups and weakly -embedded subgroups
Let be a finite group. A subgroup of is said to be weakly
S-embedded in if there exists such that is S-quasinormal
in and , where is the subgroup generated by
all those subgroups of which are S-quasinormally embedded in . We say
that is weakly -embedded in if there exists such that
is S-quasinormal in and , where
is the subgroup generated by all those subgroups of which are
-quasinormal in . In this paper, we study the properties of the weakly
S-embedded subgroups and the weakly -embedded subgroups, and use them to
determine the structure of finite groups
The Decomposition of Permutation Module for Infinite Chevalley Groups
Let be a connected reductive group defined over , the
finite field with elements. Let be an Borel subgroup defined over
. In this paper, we completely determine the composition factors
of the induced module \mathbb{M}(\op{tr})=\Bbbk{\bf G}\otimes_{\Bbbk{\bf
B}}\op{tr} (\op{tr} is the trivial -module) for any field .Comment: Accepted by Science China Mathematic
On the -norm and the --norm of a finite group
Let be a Fitting class and a formation. We call
a subgroup of a finite group
the --norm of if
is the intersection of the
normalizers of the products of the -residuals of all subgroups of
and the -radical of . Let denote a set of primes and
let denote the class of all finite -groups. We call the
subgroup of the
-norm of . A normal subgroup of is called
-hypercentral in if either or and every
-chief factor below of order divisible by at least one prime in is
-central in . Let denote the
-hypercentre of , that is, the product of all
-hypercentral normal subgroups of . In this paper, we study
the properties of the --norm, especially of the
-norm of a finite group . In particular, we investigate the
relationship between the -norm and the
-hypercentre of
On -supplemented subgroups of a finite group
A subgroup of a finite group is said to satisfy -property in
if for every chief factor of , is a
-number. A subgroup of is called to be
-supplemented in if there exists a subgroup of such that
and , where satisfies -property in . In
this paper, we investigate the structure of a finite group under the
assumption that some primary subgroups of are -supplemented in .
The main result we proved improves a large number of earlier results.Comment: arXiv admin note: text overlap with arXiv:1301.636
The Permutation Module on Flag Varieties in Cross Characteristic
Let be a connected reductive group over , the
algebraically closure of (the finite field with
elements), with the standard Frobenius map . Let be an -stable
Borel subgroup. Let be a field of characteristic . In this
paper, we completely determine the composition factors of the induced module
tr (here is the
group algebra of the group , and tr is the trivial -module). In
particular, we find a new family of infinite dimensional irreducible abstract
representations of .Comment: Accepted by Mathematische Zeitschrif
Finite groups in which SS-permutability is a transitive relation
A subgroup of a finite group is said to be SS-permutable in if
has a supplement in such that permutes with every Sylow
subgroup of . A finite group is called an SST-group if SS-permutability
is a transitive relation on the set of all subgroups of . The structure of
SST-groups is investigated in this paper
On HC-subgroups of a finite group
A subgroup of a finite group is said to be an -subgroup
of if there exists a normal subgroup of such that and for all . In this paper, we investigate the
structure of a finite group under the assumption that certain subgroups of
of arbitrary prime power order are -subgroups of
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