GROUPS AND MANY-VALUED FUNCTIONS, 327 



We have the primitive root 



y=4-i of cc'^^i (mod. lo), 

 and y is a root of the congruences 



ne'^^i (mod. 8) 

 x'^^i (mod. 4). 

 We can form 



G = {a, + c){/3, + c) (7i + c) (7.3 + c) 

 of the order 40. 



0iG= (3(«i + c)) (3(/3i + c)) (3(7i + ^+0) (3(73 + c + 3)) 

 0,Q=(g{a, + c)) (9{/3, + c)) (9(7i + c)) (9(72 + ^)) 

 03G=(27(ai + c))(27(^i + c))(27(7i + c+i))(27(73 + c + 3)) 

 which are, treating abcdefgh as 12345678, and ijklmnpq 

 as 12345678, 



G=i 234567890 a ^cf/ efg h ijklmnpq 

 2345678901 bed efg h aj k I i n p qm 

 3456789012CC? efg habklijp qm n 



©iG = 369258 x^'jocfadgbehjilkqpnm 

 69258 i^'jO'T^fadgbehcilkjpnmq 

 9258 \\']0'^badgbelicflkjinmqp 



©gG = ()'^ '] 6 ^ /\.'}^2 1 a b c d ef g h ij k I m np q 

 ^765432109 bade f g hajklinpqm 

 7654321098 cdefghabklijpqmn 



03G = 7 4i8529630c/«c?^6e Ay i Ik qp m n 

 ^i^^2()6-7)0']fadgbehcilkjpmnq 

 x^^2()6'2^o'j^adgbehcflkji7nn q p 



which is a group of the 160th order. 



And we can form with this root y = ^ sixteen equivalent 

 groups by simply varying Zy and z^^. 



46. Each of the groups J of kr substitutions above con- 

 structed is composed of a equivalent groups g^g^' -g^ each 



