Abstract
This chapter discusses the development of group theory. Group theory began with finite permutation groups. Any arrangement of n objects in a row is called a permutation of the objects. If one selects some arrangement as standard, then any other arrangement can be regarded as achieved from it by an operation of replacements. If the replacements of two operations are performed in succession, one get an arrangement that could be achieved directly by a third operation, called the product of the two operations. In a remarkable application of a group theory in its infancy, Galois showed that every algebraic equation possesses a certain permutation group on whose structure its properties depend. A generalization of the permutation group and the next step historically is the group of linear substitutions on a finite number of variables. Substitutions or matrices admit a greater freedom of algebraic treatment than do permutations.
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