## Some Properties of p-Groups and Commutative p-Groups

This article describes some properties of *p*-groups and some properties of commutative *p*-groups.

This article describes some properties of *p*-groups and some properties of commutative *p*-groups.

This article describes a rough subgroup with respect to a normal subgroup of a group, and some properties of the lower and the upper approximations in a group.

In this paper we defined the reduced residue system and proved its fundamental properties. Then we proved the basic properties of the order function. Finally, we defined the primitive root and proved its fundamental properties. Our work is based on [12], [8], and [11].

In this article, we define the set H of quaternion numbers as the set of all ordered sequences q = <*x,y,w,z*> where *x,y,w* and *z* are real numbers. The addition, difference and multiplication of the quaternion numbers are also defined. We define the real and imaginary parts of *q* and denote this by *x* = ℜ(*q*), *y* = ℑ1(*q*), *w* = ℑ2(*q*), *z* = ℑ3(*q*). We define the addition, difference, multiplication again and denote this operation by real and three imaginary parts. We define the conjugate of *q* denoted by *q*
^{*}' and the absolute value of *q* denoted by |q|. We also give some properties of quaternion numbers.

This article introduces definitions of circulant matrices, line-and column-circulant matrices as well as anti-circular matrices and describes their main properties.

MML identifier: MATRIX16, version: 7.9.03 4.108.1028

In this article, we give some important theorems of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions.

This article describes definitions of subsymmetric matrix, anti-subsymmetric matrix, central symmetric matrix, symmetry circulant matrix and their basic properties.

This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arctan and arccot.

MML identifier: SIN COS9, version: 7.8.10 4.100.1011

In this article, we give several integrability formulas of special and composite functions including trigonometric function, inverse trigonometric function, hyperbolic function and logarithmic function.

In this article, we give several differentiation and integrability formulas of special and composite functions including trigonometric function, and polynomial function.