The origin of group theory can be traced back to19th century, and mathematicians began to study symmetry and transformation groups. In this process, they found some mathematical structures with the same properties, which can be expressed by groups. With the passage of time, group theory has gained more and more attention and development, and has become an important part of modern mathematics.
The basic concepts of group theory include the definition of group, subgroup, homomorphism, isomorphism and so on. A group is defined as above. A subgroup refers to a subset of a group, which is itself a group. Homomorphism refers to the mapping between two groups, which keeps the operation of groups. Isomorphism means that there is a bijection between two groups, and this mapping is homomorphic. Isomorphic groups are called isomorphic classes, and they are similar in structure and properties.
The research methods of group theory mainly include classification and construction. Classification refers to the classification of groups into different categories according to their properties and structures, such as finite groups, infinite cyclic groups, commutative groups and so on. Construction refers to obtaining new groups by some means on the basis of known groups. This method is very useful in solving some mathematical problems, because it can help us find new information related to known problems.
In a word, mathematical group theory is a subject that studies abstract algebraic structure, with emphasis on groups and their properties. Group theory is widely used in mathematics, which provides a powerful tool for understanding many complex mathematical phenomena. With the development of mathematics, group theory will continue to play its important role and promote the progress in the field of mathematics.