The set of all (n×n) non-singular matrices over the field F. And this set forms a group under the operation of matrix multiplication. This group is called the general linear group of dimension  over the field F, denoted by . The determinant of these matrices is a homomorphism from  into F* and the kernel of this homomorphism was the special linear group and denoted by  Thus  is the subgroup of  which contains all matrices of determinant one.

The rationally valued characters of the rational representations are written as a linear combination of the induced characters for the groups discussed in this paper. We find the Artin indicator for this group after studying the rationally valued characters of the rational

The concept of the order sum graph associated with a finite group based on the order of the group and order of group elements is introduced. Some of the properties and characteristics such as size, chromatic number, domination number, diameter, circumference, independence number, clique number, vertex connectivity, spectra, and Laplacian spectra of the order sum graph are determined. Characterizations of the order sum graph to be complete, perfect, etc. are also obtained.

The set of all (n×n) non-singular matrices over the field F this set forms a group under the operation of matrix multiplication. This group is called the general linear group of dimension  over the field F, denoted by . The determinant of these matrices is a homomorphism from  into F* and the kernel of this homomorphism was the special linear group and denoted by  Thus  is the subgroup of  which contains all matrices of determinant one.

The rational valued characters of the rational representations written as a linear combination of the induced characters for the groups  discuss in this paper and find the Artin indicator for this group after study the rational valued characters of the rational representations and the induce

This paper reflects the evidence of the Islamic Development Bank's role in building human and institutional capacities in Member States and assisting governments in achieving economic development. The Bank's activities in the area of ​​capacity development are an important part in following up the economic developments, And help governments to build and formulate effective policies and institutions to implement more effective macro policies leading to better economic outcomes. It is therefore helpful to help Member States increase their resources, modernize their banking systems, establish effective legal frameworks and improve Reports on macroeconomic and financial statistics. The Bank has faced many challenges and risks tha

The main objective of this paper is to find the order and its exponent, the general form of all conjugacy classes, Artin characters table and Artin exponent for the group of lower unitriangular matrices L(3,? p ), where  p  is prime number.

Background: Tooth wear is one of the most concerning problems of the current dental practice especially among older subjects. The aim of this study is to determine the severity of tooth wear and its relation with selected salivary variables (salivary pH and vitamin C level) among a group of older adults in Mosul city/Iraq. Materials and methods: All subjects (30 subjects) of both gender tookpart in the current study; sixteen of them were older adults (55-65 years) and compared with fourteen middle-aged adults (30-40 years) at Textile factory in Mosul city/Iraq. Unstimulated salivary samples were collected and salivary pH was immediately measured. Salivary vitamin C level was determined colormetrically. Severity of tooth wear was determined