Let $G = \left \langle g \right \rangle$ be a cyclic group of order 20

Find all generators of the subgroup of order 10?

At this point, I refer to the

corollary: Let $G = \left \langle g \right \rangle$ be a cyclic group of order n. Then $G = \left \langle a^{k} \right \rangle$ if and only if$ gcd\left ( n,k \right )=1$

The hint I am given is:

The generator of H are all elements $\left ( g^{2} \right )^{i}$

And I can proceed henceforth. But why is the subgroup H = $\left \langle g^{2} \right \rangle$ the unique subgroup? A little nudge would help.


Assume $h \in \langle h\rangle \subset H$ where $h=g^k$ and $k$ is not even. Then $k$ is a multiple of 5 because if not it would be coprime to $20$ and hence generate $G$. But being a multiple of 5, it has order 4 (not 2 because it would be even then) inparticular it cannot be an element of $H$.


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