Number of Powers of Cyclic Group Element
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Theorem
Let $G$ be a cyclic group of order $n$, generated by $g$.
Let $d \divides n$.
Then the element $g^{n/d}$ has $d$ distinct powers.
Proof
Follows directly from Order of Subgroup of Cyclic Group:
- $\order {\gen {g^{n/d} } } = \dfrac n {\gcd \set {n, n/d} } = d$
Thus from List of Elements in Finite Cyclic Group:
- $\gen {g^{n/d} } = \set {e, g^{n/d}, \paren {g^{n/d} }^2, \ldots, \paren {g^{n/d} }^{d - 1} }$
and the result follows.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Proposition $4.14$ (passim)