Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn finite totally 2-closed groups
68
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
An abstract group $G$ is called totally 2-closed if $H = H^{(2),Omega}$ for any set $Omega$ with $Gcong Hleqtextrm{Sym}_Omega$, where $H^{(2),Omega}$ is the largest subgroup of symmetric group of $Omega$ whose orbits on $OmegatimesOmega$ are the same orbits of $H$. In this paper, we prove that the Fitting subgroup of a totally 2-closed group is a totally 2-closed group. We also conjecture that a finite group $G$ is totally 2-closed if and only if it is cyclic or a direct product of a cyclic group of odd order with a generalized quaternion group. We prove the conjecture in the soluble case, and reduce the general case to groups $G$ of shape $Zcdot X$, with $Z = Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian
rate research
Read More
We show that groups presented by inverse-closed finite convergent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to the free product of finitely many finite groups and copies of $mathbb Z$) if and only if a certain relation on the set of non-trivial finite-order elements of the group is transitive on a bounded set. We use this to prove that deciding if a group presented by an inverse-closed finite convergent length-reducing rewriting system is not plain is in $mathsf{NP}$. A yes answer would disprove a longstanding conjecture of Madlener and Otto from 1987. We also prove that the isomorphism problem for plain groups presented by inverse-closed finite convergent length-reducing rewriting systems is in $mathsf{PSPACE}$.
In this paper, we study a group in which every 2-maximal subgroup is a Hall subgroup.
A theorem of Dolfi, Herzog, Kaplan, and Lev cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in cite{DHKL}, we establish the following generalisation: if $mathfrak{X}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $overline{mathfrak{X}}$ is the extension-closure of $mathfrak{X}$, then there exists an (optimal) constant $gamma$ depending only on $mathfrak{X}$ such that, for all non-trivial finite groups $G$ with trivial $mathfrak{X}$-radical, begin{equation} leftlvert G^{overline{mathfrak{X}}}rightrvert ,>, vert Gvert^gamma, end{equation} where $G^{overline{mathfrak{X}}}$ is the ${overline{mathfrak{X}}}$-residual of $G$. When $mathfrak{X} = mathfrak{N}$, the class of finite nilpotent groups, it follows that $overline{mathfrak{X}} = mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J.,G. Thompsons classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations $mathfrak{X}$ of full characteristic such that $mathfrak{S} subset overline{mathfrak{X}} subset mathfrak{E}$, thus providing applications of our main result beyond the reach of cite[Thm.~C]{DHKL}.
In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings $R$, in particular for $R = mathbb{Z}$ and $R= mathbb{Q}$. We show these properties satisfy many analogous results to the case of discrete groups, and we provide analogues of the famous Bieris and Browns criteria for finiteness properties and deduce that both $FP_n$-properties and $F_n$-properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness properties.
A finite group $G$ is called a Schur group, if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this paper it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any non-cyclic abelian Schur group of odd order is isomorphic to $Z_3times Z_{3^k}$ or $Z_3times Z_3times Z_p$ where $kge 1$ and $p$ is a prime. In addition, we prove that $Z_2times Z_2times Z_p$ is a Schur group for every prime $p$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
