Amount Of Generators For Cyclic Group And Euler's Function
Answer : A cyclic group with n n n elements has φ ( n ) \varphi(n) φ ( n ) generators. It seems like what you're getting confused by is that: The group Z 7 × = { 1 , 2 , 3 , 4 , 5 , 6 } \mathbb{Z}_7^\times=\{1,2,3,4,5,6\} Z 7 × = { 1 , 2 , 3 , 4 , 5 , 6 } is a cyclic group with 6 elements , and therefore it should considered equivalent to (more precisely, it is isomorphic to) the cyclic group Z 6 \mathbb{Z}_6 Z 6 , and you should expect φ ( 6 ) = 2 \varphi(6)=2 φ ( 6 ) = 2 generators, not φ ( 7 ) = 6 \varphi(7)=6 φ ( 7 ) = 6 generators The elements 4 4 4 and 6 6 6 in Z 7 × \mathbb{Z}_7^\times Z 7 × are not generators: the powers of 4 4 4 in Z 7 × \mathbb{Z}_7^\times Z 7 × are { 1 , 4 , 2 } \{1,4,2\} { 1 , 4 , 2 } , and the powers of 6 6 6 in Z 7 × \mathbb{Z}_7^\times Z 7 × are { 1 , 6 } \{1,6\} { 1 , 6 } . (The only generators of Z 7 × \mathbb{Z}_7^\times Z 7 × are 3 3 3 and 5 5 5 , and there are exactly 2 = φ ( 6 ) 2=\varphi(6) 2 = φ ( 6 ) of th...