GROUPS AND MANY-VALUED FUNCTIONS. 395 



34217865 

 65781243 

 56872134 

 87653412 

 78564321 



equivalent to the preceding, which has sis more square 

 roots of 21436587. 



What has just been done is a case of a more general 

 theorem on grouped groups. The two following 

 1234 1234 



2*^1 4 3*^ 2*^1 4'^3 

 3'^4'^i 2 3*^4 1 ^'^ 



4 3*^2 i**, 4*^3*^2 \, 

 are true groups, if we define that 



1 _i, 2 —2, 3 —3, 4 —4, 

 and that no substitution is changed in value by having its 

 four exponents increased each by the same number. For 

 example : 



3*^4*^ 1 2 • 43*^2 1 '' = 2 1 '^4''3^'^~^ =: 2 1 '^4''3 = 2*^ 1 43"^. 

 We may write in either of these groups, for v, any 

 square group u of powers of a substitution of the (2c?- 2)"' 

 order made with id-i elements, v being each one of the 

 four elements 1234, and for v^ the result oi d- 1 cyclical 

 permutations of the vertical rows of u. 



For example : if d=. 4, we have by the latter of the two 

 auxiliaries the group following : 



123456 'i%(^oah cdefgh ijklmn 

 234561 Sgoaby defghc jklmni 

 345612 goabyS efghcd klmnij 

 456123 oabySg fghcde Imnijk 

 561234 067890 ghcdef mnijkl 

 612345 67890a hgcdef nijklm 



oabySg 123456 Imnijk cdefgh 

 ahySgo 234561 mnijkl defghc 



