Page:Popular Science Monthly Volume 64.djvu/374

which he implicitly admits and which he does not deem even necessary to enunciate. This is tantamount to saying that the former are the fruit of later experience, that the others were first assimilated by us, and that consequently the notion of group existed prior to all others. . . . What we call geometry is nothing but the study of formal properties of a certain continuous group, so that we may say space is a group.^[1]

From these words of Poincarế it follows that the group concept is implicitly involved in some of the earliest mathematical developments. In an explicit form it first appears in the writings of Lagrange and Vandermonde in 1770. These men inaugurated a classic period in the theory of algebraic equations by considering the number of values which a rational integral function assumes when its elements are permuted in every possible manner. For instance, if the elements of the expression $ab+cd$ are permuted in every possible manner, it will always assume one of the following three values: $ab+cd$, $ac+bd$, $ad+bc$.

The eight different permutations which do not change the value of one of these expressions are said to form a permutation group and the expression is said to belong to this group. There is always an infinite number of distinct expressions which belong to the same permutation group. Hence it is convenient for many purposes to deal with the permutation group rather than with the expressions themselves. This fact was recognized very early and led to the study of permutation groups, especially in connection with the theory of algebraic equations. The most fundamental work along this line was done by Galois, who influenced the later development most powerfully, although he died when only twenty years old.

Galois first proved (about 1830) that the solution of any given algebraic equation depends upon the structure of the permutation group to which the equation belongs. As the algebraic solution of equations occupies such a prominent place in the history of mathematics this discovery of Galois furnished a powerful incentive for the study of permutation groups. Before Galois an Italian named Ruffini and a Norwegian named Abel had employed permutation groups to prove that the general equation of the fifth degree can not be solved by successive extraction of roots. In doing this the former studied a number of properties of permutation groups and is therefore generally regarded as the founder of this theory.

The definition of a permutation group is very simple. It is merely the totality of distinct permutations which do not change the formal value of a given expression. Such a totality of permutations has many remarkable properties. One of the most important of these is the fact that any two of them are equivalent to some one. That is, if

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1. Poincaré, The Monist, vol. 9, 1898, pp. 34 and 41.