Problem :
- Find all abelian groups of order 72, up to isomorphism.
- Which one of those groups are cyclic?
- Amont these subgroups, find all groups with a unique subgroup of order 4.
My answer : Let $G$ a abelian group such that $|G|=72=2^33^2$. We can find the elementary divisors of $G$ : $(2^3,3^2) ; (2,2^2,3^2) ; (2,2,2,3^2) ; (2^3,3,3);(2,2^2,3,3);(2,2,2,3,3)$.
It follows that $G$ is isomorphic to 6 abelian groups of order 72: $$C_{72}$$ $$C_2\times C_{36},C_2\times C_2\times C_2$$ $$C_{24}\times C_3, C_6\times C_{12}, C_2\times C_6\times C_6$$
Now I can't seem to find which groups are cyclic. Quite intuitively I would say $C_{72}$ is cyclic because generated by $1$, $C_2\times C_{36},C_2\times C_2\times C_2$ is generated by $(1,1,1)$ but I don't have a rigorous answer.
For the subgroup of order 4, I know that if $G$ is a cyclic group of order $m$ and if $n|m$, then there exists a unique subgroup of $G$ of order $n$. Then the group $C_{72}$ has a unique subgroup of order $4$ but what about the others?