GROUPS AND MANY-VALUED FUNCTIONS. 275 



2, It is sufficient to consider groups of which an ar- 

 rangement is the natural order i 2 3' -N of the elements, 

 which is called unity, and may be denoted by (1). Such 

 a group is 



Every substitution in this group may be written with 

 unity for denominator ; that is, we have always 



A A 



A. -"(7)' 

 A,, being one of the k arrangements. And the k substi- 

 tutions of the group are 



(1) Ai A2 -c^k-i 



which may be written 



(ijAiAa' 'h.jc_i. 

 A group of k permutations, of which one is unity, is a 

 model group of the order k. 



The simplest definition of a group is this — that the 

 product of any two substitutions of the group is a substi- 

 tution of the group ; that is, in any model group, 



Ap Ag = A^ 



A A _ A2_ A 



A^ and A^ being in the group. It follows that every power 

 of a substitution is in the group. 

 If A^=i, 



we say that A^ is a substitution of the U^ order. We have 

 also 



A6-C — A—" • 



that is, every negative power of any substitution is one of 

 the group. In fact, 



Aj, 1 1 A^, 



1 -A|-A,- 1- 



3. It is important to be able to form readily the pro- 

 duct of two substitutions A^ and A,^ in a model group. 

 The simple rule is : Pronounce the arrangement Ay^ '^^^^ 



