Finitely generated abelian groups

Show that in a finite abelian group $A$, for any $d|| A \mid$ there exists at least one subgroup of order $d$ (the converse of Lagrange's theorem).

Show that, with a suitable ordering, the invariants of any subgroup are divisors of the invariants of the abelian group.

If $A \oplus A \cong B \oplus B$, where $A$ and $B$ are finite abelian groups, then $A \cong B$.

If $A, B, C$ are finite abelian groups and $A \oplus C \cong B \oplus C$, then $A \cong B$.

Every finite abelian group of order $n$, not divisible by the square of an integer $>1$, is cyclic.

List all the nonisomorphic abelian groups of order 72.

Are the groups $Z_{12} \oplus Z_{72}$ and $Z_{18} \oplus Z_{48}$ isomorphic?

State and prove the theorem on finitely generated abelian groups in the language of integer matrices (matrices with coefficients in $\mathbb {Z}$). Use appropriately elementary transformations on such matrices.

Prove that the index of a subgroup $A \subseteq F_{n}^{a b}$ of the free abelian group $F_{n}^{a b}$ of rank $n$ is finite if and only if $\operatorname {rank} A=n$.