WHAT IS GROUP THEORY? 37 1 



any one of these permutations is repeated, or is followed by some other 

 permutation in the totality, the result is equivalent to a single permu- 

 tation in the totality. This property is characteristic; for if any set 

 of distinct permutations possesses this property they form a permuta- 

 tion group and it is possible to construct an infinite number of expres- 

 sions such that they are unchanged by these permutations but by no 

 others. 



Soon after the fundamental properties of permutation groups be- 

 came known, it was observed that many other operations possess the 

 same properties. This gradually led to more abstract definitions of 

 the term group. According to the earliest of these any set of distinct 

 operations such that no additional operation is obtained by repeating 

 one of them or combining any two of them was called a group. All 

 the later definitions included this property, but they generally add 

 other conditions. These additional conditions are frequently satisfied 

 by the nature of the operations which are under consideration and 

 hence do not always require attention. This may account for the 

 fact that the oldest definition is still very commonly met in text-books, 

 notwithstanding the fact that the ablest writers on the subject aban- 

 doned it a long time ago. 



The three additional conditions which a set of distinct operations 

 must satisfy in order that it becomes a group when the operations 

 are combined are: (1) The associative law must be satisfied; i. e., if 

 r, s, t represent any three operations of the set, then the three suc- 

 cessive operations rst must give the same result independently of the 

 fact whether we replace rs or st by a single operation. The operations 

 are, however, not generally commutative, that is, rs may be different 

 from sr. (2) From each of the two equations rs=^ts, sr^=st it 

 follows that r = t. (3) If the equation xy = z involves two opera- 

 tions of the set the third element of the equation must also represent 

 an operation in the set. It may be observed that the totality of 

 integers combined by multiplication obey all these conditions except 

 the last. Hence this totality does not form a group with respect to 

 multiplication, although the contrary has frequently been affirmed.* 



One of the simplest instances of a group of operations is furnished 

 by the n different numbers which satisfy the equation a;"= 1. It is 

 very easy to see that these numbers obey each of the four given condi- 

 tions when they are combined by multiplication. Hence we say that 

 the n roots of the equation a;"= 1 form a group with respect to multi- 

 plication. Since all these roots are powers of a single one of them 



* Among other places this error is found in the first edition of Weber's 

 classic work on algebra, vol. 2, p. 54. It has been corrected in the second 

 edition of this work. Somewhat simpler definitions of the term group have 

 recently been given by Huntington and Moore, Bulletin of the American Mathe- 

 matical Society, vol. 8, p. 388. 



