$G \cong \mathbb{Z}/3^6\mathbb{Z} \oplus\mathbb{Z}/3^5\mathbb{Z}\oplus\mathbb{Z}/3^2\mathbb{Z}$
$H\leq G$ so that $G/H \cong \mathbb{Z}/3^2\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z} $
Find all possible $H$ up to isomorphism.
This is what I did:
(1) $H$ must be of the form $\mathbb{Z}/3^{k_{1}}\mathbb{Z} \oplus\mathbb{Z}/3^{k_{2}}\mathbb{Z}\oplus\mathbb{Z}/3^{k_{3}}\mathbb{Z}$
(2) WLOG, $0 \leq k_{1} \leq 6$, $0 \leq k_{2} \leq 5$, $0 \leq k_{3} \leq 2$, (The order obiously doesn't matter because a change of order will be isomorphic)
Also, $|G/H|=3^3=3^{13}/(3^{{k_{1}+k_{2}+k_{3}}}) \Rightarrow k_{1}+k_{2}+k_{3}=10$
Thus, by choosing $k_{i}$ we can find all possibilities.
However, I have a few questions:
- Is (1) justified? Why $\mathbb{Z}/3^{k_{1}}\mathbb{Z} \oplus\mathbb{Z}/3^{k_{2}}\mathbb{Z}\oplus\mathbb{Z}/3^{k_{3}}\mathbb{Z}\oplus\mathbb{Z}/3^{k_{4}}\mathbb{Z}$ isn't possible?
- Is (2) justified? Why $k_{1}=7$ is not possible?
Thank you!