Evaluation of All Subgroups of a Group of Higher Order 600, 650 and 700 By Using Sylow’s Theorem

Abstract

We develop the all possible subgroups of a group of higher order 600, 650 and 700 of different algebraic structures as groups by using Sylow’s theorem. In fact, the help of order of group(o(G)) subgroups, homomorphism, isomorphism, dihedral group, and split extensions of groups are used in Sylow’s theorems. First of all, we discuss the order of a group and the order of elements of a group in real numbers. Then we develop all the applications of the Sylow’s theorems of a group in higher order of a group which will give the knowledge of the mathematical systems like number systems, vectors, matrices and group theory and so on. When we study Sylow’s theorem of higher order of a group, then we use short exact sequences, and split extensions. The number of abelian groups is calculated by the product of prime powers or other cases but the non-abelian case is dependent on a different theorem. The use of two cases like abelian group and non-abelian group when the study of the Sylow’s theorems of a group in higher order of a group. After that we find out the abelian group and non-abelian group in the higher order of the group by using Sylow’s theorem.