All infinite cyclic groups have exactly generators. To see this, let be an infinite cyclic group and be a generator of . Let such that is a generator of . Then . Then . Thus, there exists with . Therefore, . Since is infinite and must be infinity, . Since and and are integers, either or . It follows that the only generators of are and .
A finite cyclic group of order has exactly generators, where is the Euler totient function. To see this, let be a finite cyclic group of order and be a generator of . Then . Let such that is a generator of . By the division algorithm, there exist with such that . Thus, . Since is a generator of , it must be the case that . Thus, . Therefore, , and the result follows.
|Date of creation||2013-03-22 13:30:39|
|Last modified on||2013-03-22 13:30:39|
|Last modified by||Wkbj79 (1863)|