314 MILLER— SUBSTITUTION GROUP PROPERTIES [June 4, 



augmented co-sets may be so named respectively that there results 

 the same substitution group when the left augmented co-sets are 

 multiplied on the left as when the right augmented co-sets are 

 multiplied on the right. When H is invariant under G K is regular, 

 and iv and K" are each composed of all the substitutions on these 

 letters which are separately commutative with every substitution of 

 the other, K" being obtained by left-hand multiplication when the 

 co-sets (which reduce to single operators in this case) are named 

 the same as in the right co-set. As K" reduces to K when these 

 co-sets are named in the manner noted above it results from what 

 has been proved that K and K" are conjugate. When K is com- 

 posed of substitutions of order two in addition to the identity, the 

 two given methods of naming the co-sets will coincide and hence 

 the given process also gives, as a special case, a proof of the 

 theorem that every group in which all the substitutions besides the 

 identity are of order 2 must be abelian. It should be added that the 

 main results of this section are not new but the subject is so im- 

 portant that these details should be of some interest as they throw 

 new light on the entire process. 



University of Illinois, 

 May, 1910. 



