Number of Generators of Cyclic Group whose Order is Power of 2

Theorem

Let $G$ be a finite cyclic group.

Let the order of $G$ be $2^k$ for some $k \in \Z_{>0}$.

Then $G$ has $2^{n - 1}$ distinct generators.

Proof

From Finite Cyclic Group has Euler Phi Generators, $G$ has $\map \phi {2^n}$ generators.

The result follows from corollary to Euler Phi Function of Prime Power.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $19$