In the space of marked GROUPs, we determine the structure of GROUPs which are limit points of the set of all GENERALIZED QUATERNION GROUPs.

Bentea and Tarnauceanu (An. Stiint. Univ. Al. I. Cuza Ias, Ser. Noua, Mat., 54 (1) (2008), 209-220) proposed the following problem: Find an explicit formula for the number of fuzzy subGROUPs of a finite hamiltonian GROUP of type Q8´Zn where Q8 is the QUATERNION GROUP of order 8 and n is an arbitrary odd integer. In this paper we consider more general GROUP: the direct product of a GENERALIZED QUATERNION GROUP of any even order and a cyclic GROUP of any odd order. For this GROUP we give an explicit formula for the number of fuzzy subGROUPs.

In this paper, we compute the number of fuzzy subGROUPs of some classes of non-abeilan GROUPs. Explicit formulas are given for dihedral GROUPs D2n, quasi-dihedral GROUPs QD2n, GENERALIZED QUATERNION GROUPs Q4n and modular p-GROUPs Mpn.

Let R be a commutative ring with unity of characteristic r≥ 0 and G be a locally finite GROUP. For each x and y in the GROUP ring RG define [x, y]=xy-yx and inductively via [x, _( n+1) y]=[[x, _( n) y], y]. In this paper we show that necessary and sufficient conditions for RG to satisfies [x^m(x, y), _( n(x, y)) y]=0 is: 1) if r is a power of a prime p, then G is a locally nilpotent GROUP and G' is a p-GROUP, 2) if r=0 or r is not a power of a prime, then G is abelian. In this paper, also, we define some GENERALIZED Engel conditions on GROUPs, then we present a result about unit GROUP of GROUP algebras which satisfies this kind of GENERALIZED Engel conditions...

Let (R; m) be a commutative noetherian local ring and 􀀀 be a  nite GROUP. It is proved that if R admits a dualiz-ing module, then the GROUP ring R􀀀 has a dualizing bimodule, as well. Moreover, it is shown that a  nitely generated R􀀀-module M has GENERALIZED Gorenstein dimension zero if and only if it has GENERALIZED Gorenstein dimension zero as an R-module.

In this paper, we dene the classes F (p, q, s) of QUATERNION-valued functions, then we characterize QUATERNION Bloch functions by QUATERNION F (p, q, s) functions in the unit ball of R3. Further, some important basic properties of these functions are also considered.

Inverse Young inequality asserts that if n>1, then|zw|³n|z|1/n+(1-n)|w|1/1-nfor all complex numbers z and w, and equality holds if and only if|z|1/n=|w|1/1-n..In this paper the complex representation of QUATERNION matrices is applied to establish the inverse Young inequality for matrices of QUATERNIONs. Moreover, a necessary and sufficient condition for equality is given.

This article uses impulsive control along with QUATERNION parameters instead of Euler angles in kinematics equations of satellite. The QUATERNION parameters are applied to overcome singularity problem in the numerical solution. It is assumed that the satellite is subjected to deterministic external perturbations. At  rst, the chaotic behavior of system is investigated when there is no control on the system. Then, impulsive control is used to stabilize the satellite attitude around the equilibrium point of origin. Finally, simulation results are given to visualize the e ectiveness and feasibility of the proposed method.