Number of Powers of Cyclic Group Element

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)