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 Historical Note:


There are three historical roots of the development of abstract group theory evident in the mathematical literature of the nineteenth century: the theory of algebraic equations, number theory and geometry. All three of these areas used group theoretic methods of reasoning, although the methods were considerably more explicit in the first area than in the two.

                    One of the central themes of geometry in the nineteenth century was the search of invariants under various types geometric transformations. Gradually attention became focused on the transformations themselves, which in many cases can be thought of as elements of groups. 

      In number theory, already in the eighteenth century Leonhard Euler had considered the remainders on division of power 𝑎𝑛 by fixed prime 𝑝. These remainders have “group” properties. Similarly, Carl F. Gauss, 

Finally, the theory of algebraic equations provided the most explicit prefiguring of the group concept. 

Joseph-Louis Lagrange (1736 − 1813) in fact initiated the study of permutations of the roots of an equation as a tool for solving it. These permutations, of course, were ultimately considered as elements of a group.

It was Walter von Dyck (1856 − 1934) and Heinrich Weber (1842 − 1913) who is 1882 were able independently to combine the three roots and give clear definitions of the notion of an abstract group.

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