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In this study we have investigated both the discipline of pure mathematics and its application to the solution of significant mathematical problems. In mathematics, we frequently utilise apparatus designed for multiple purposes. Contemporary algebra refers to the study of groups as group theory. In this context, a group is a device that satisfies a set of axioms by joining together to satisfy a fixed set of axioms and is made up of a fixed set of devices and binary characteristics that can be used in units. This requires that the Groups be shut down as part of the procedure, that it adheres to the merger rule, and that it consists of an inverted proprietary object. When a set also satisfies the rule of exchange, it is known as an abelian or versatile group. This is the case because the set conforms to the law of exchange. An abelian group is a collection of subscript numbers in which the detail equals zero and the alternative may be of either high or low argument variety quality. Groups play a crucial role in modern algebra; their fundamental structure can be observed in the most radical mathematical situations. In geometry, the concept of groups can be observed; these groups characterise activities that occur in conjunction with measurements and particular transformations. The group concept can be applied to disciplines such as physics, chemistry, and computer science.