We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component Mandelbrot set can be conformally mated a congruence subgroup of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper P normal S L left-parenthesi...

The object of this note is to give a very short proof of the following theorem of Ivanov and Schupp. Let H be a finitely generated subgroup of a free group F and the index [F : H ] infinite. Then there exists a nontrivial normal subgroup N of F such that N ∩ H = {1}. As noted in the abstract above the object of this note is to give a new proof of the following theorem of S.V.Ivanov and P.E.Schu...

In this paper, we introduce some properties of an intuitionistic normal fuzzy m-subgroup of m- group with m-homomorphism and isomorphism. We study he image, the pre-image and the inverse mapping of the intuitionistic normal fuzzy m-‎subgroups.‎

a normal subgroup $n$ of a group $g$ is said to be an‎ ‎$emph{omissible}$ subgroup of $g$ if it has the following property‎: ‎whenever $xleq g$ is such that $g=xn$‎, ‎then $g=x$‎. ‎in this note we construct various groups $g$‎, ‎each of which has an omissible subgroup $nneq 1$ such that $g/ncong sl_2(k)$ where $k$ is a field of positive characteristic‎.

we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.‎

A subgroup of V containing a principal congruence subgroup T(«) is said to be a congruence subgroup, and is of level « if « is the least such integer. In a recent article [2] the writer determined all normal subgroups of T of genus 1 (see [l] for the definition of the genus of a subgroup of r). An interesting question that arises is to decide which of these are also congruence subgroups. In thi...

Let Γ be a finitely presentable pro-p group such that def(Γ) > 0. If Γ has a nontrivial finitely generated closed normal subgroup N of infinite index then def(Γ) = 1, N is a free pro-p group and Γ/N is virtually free. If π is a finitely presentable group with β (2) 1 (π) = 0 then def(π) ≤ 1, with equality only if c.d.π = 2 or π ∼= Z. (See Theorem 2.5 of [1].) The L-Betti number condition holds ...

We say that a countable discrete group Γ satisfies the invariant von Neumann subalgebras rigidity (ISR) property if every Γ- subalgebra M in L(Γ) is of form L(Λ) for some normal subgroup Λ◁Γ. show many “negatively curved” groups, including all torsion free non-amenable hyperbolic groups and with positive first L2-Betti number under mild assumption, certain finite direct product them have this p...