370 POPULAR SCIENCE MONTHLY. 



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.* 



From these words of Poincare it follows that the group concept 

 is implicitly involved in some of the earliest mathematical develop- 

 ments. 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 db + cd are permuted in every possible manner, it 

 will always assume one of the following three values : ah -\- cd, ac -\- bd, 

 ad + be. 



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 per- 

 mutation group. Hence it is convenient for many purposes to deal 

 with the permutation group rather than with the expressions them- 

 selves. 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 mathe- 

 matics 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 gen- 

 erally 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 



* Poincare, The Monist, vol. 9, 1898, pp. 34 and 41. 



