we derive the formulas of the maximal andminimal ranks of four real matrices $x_{1},x_{2},x_{3}$ and $x_{4}$in common solution $x=x_{1}+x_{2}i+x_{3}j+x_{4}k$ to quaternionmatrix equations $a_{1}x=c_{1},xb_{2}=c_{2},a_{3}xb_{3}=c_{3}$. asapplications, we establish necessary and sufficient conditions forthe existence of the common real and complex solutions to the matrixequations. we give the exp...

Quaternion-valued signals along with quaternion Fourier transforms (QFT) provide an effective framework for vector-valued signal and image processing. However, the sampling theory of quaternion valued signals has not been well developed. In this paper, we present the generalized sampling expansions associated with QFT by using the generalized translation and convolution. We show that a σ-bandli...

In this paper, we compute the number of fuzzy subgroups of some classes of non-abeilan groups. Explicit formulas are givenfor dihedral groups $D_{2n}$, quasi-dihedral groups $QD_{2^n}$, generalized quaternion groups $Q_{4n}$ and modular $p$-groups $M_{p^n}$.

Suppose that a group G acts transitively on the points of a nonDesarguesian plane, P. We prove first that the Sylow 2-subgroups of G are cyclic or generalized quaternion. We also prove that P must admit an odd order automorphism group which acts transitively on the set of points of P. 1 MSC(2000): 20B25, 51A35.

Graph theory can give a representation of abstract mathematical systems such as groups or rings. We have many graph representations for group, in this study we use the coprime generalized quaternion group to find numerical invariants graph, which are clique number and chromatic number. The main results obtained from is equal each case order.

Let $G$ be a finite group. The reduced power graph of is the undirected whose vertex set $G$, and two distinct vertices $x$ $y$ are adjacent if $\langle x \rangle \subset \langle y \rangle$ or \rangle$. In this paper, we give tight upper lower bounds for metric dimension As applications, compute $\mathcal{P}$-group, cyclic group, dihedral generalized quaternion group odd order.