Introduction to Group Theory

The theory of groups, an important part in present day mathematics, started early in nineteenth century in connection with the solutions of algebraic equations. Originally a group was the set of all permutations of the roots of an algebraic equation which has the property that combination of any two of these permutations again belongs to the set. Later the idea was generalized to the concept of an abstract group. An abstract group is essentially the study of a set with an operation defined on it. Group theory has many useful applications both within and outside mathematics. Groups arise in a number of apparently unconnected subjects. In fact they appear in crystallography and quantum mechanics in geometry and topology, in analysis and algebra and even in biology. Before we start talking of a group it will be fruitful to discuss the binary operation on a set because these are sets on whose elements algebraic operations can be made. We can obtain a third element of the set by combining two elements of a set. It is not true always. That is why this concept needs attention.