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If is a cyclic group and , then is a generator of if
.
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.
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