in this paper, we study the notion of solvable $l$-subgroup of an $l$-group and provide its level subset characterization and this justifies the suitability of this extension. throughout this work, we have used normality of an $l$-subgroup of an $l$-group in the sense of wu rather than liu.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

Following Keating and Rudnick, we say that a character Λ : B× → C× is “even” if it is trivial on the subgroup k×. The quotient group B×/k× is the group of “big” Witt vectors mod X with values in k. Recall that for any ringA, the groupBigWitt(A) is simply the abelian group 1 +XA[[X]] of formal series with constant term 1, under multiplication of formal series. In this group, the elements 1 +XA[[...

We study the definability of certain subgroups of a group G that does not have the independence property. If a (type) definable subset X of an elementary extension G of G has property P , we call its trace X∩G over G an externally (type) definable P set. We show the following. Centralisers of subsets of G are externally definable subgroups. Cores of externally definable subgroups and iterated c...

A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size soluble residual is bounded from below by certain power order, inequality sharp. Inspired this result some arguments \cite{DHKL}, we establish following generalisation: if $\mathfrak{X}$ subgroup-closed formation full characteristic which does not contain all gr...

We show in this paper that for any group G of permutations of an (n + 1) set, any composite q > (n + 1)’ is the order of an n quasigroup Q whose conjugate invariant subgroup is precisely G. If y1 is a positive integer, and X is a finite set of cardinality IX/ = q, we say that Q is an n quasigroup on X of order q if Q is a (4”) X (n $ 1) array, with entries from X, with the property that each of...

Let O be a henselian discrete valuation ring with perfect residue field, G a connected reductive group over k, the quotient field of O. Let S be a maximal k-split torus of G, = (G, S) the corresponding root system. For simplicity, in this introduction we assume that (G, S) is reduced. Fix a point x on the apartment A(G, S) attached to S. According to Bruhat-Tits theory, x determines a filtratio...

Lattices and parabolic subgroups are the obvious examples of cocompact subgroups of a connected, semisimple Lie group with finite center. We use an argument of C. C. Moore to show that every cocompact subgroup is, roughly speaking, a combination of these. We study a cocompact subgroup H of a connected, semisimple Lie group G with finite center. The case where H is discrete is very important and...

The Mackey–Zimmer theorem classifies ergodic group extensions $X$ of a measure-preserving system $Y$ by compact $K$, showing that such are isomorphic to skew-product $X \equiv Y \rtimes _\rho H$ for some closed subgroup $

We are concerned with torsion subgroups of Lie groups. We extend the classical result of C. Jordan on the structure of finite linear groups to torsion subgroups of connected Lie groups. 1. In [1], Boothby and Wang proved that, for any connected Lie group G, there exists a number k{G) such that any finite subgroup contains an abelian normal subgroup whose index is bounded by k{G), thereby genera...