We know that by Fundamental theorem of Finite abelian groups, any Finite abelian group $A$ can be expressed as $A\cong \langle a_1\rangle \times \langle a_2 \rangle\times \cdots \times \langle a_n \rangle$.
But do these cyclic factors intersect trivially? Can we say that $\langle a_i \rangle \cap \langle a_j \rangle=e$ for $i\neq j$. If it is can someone explain it please.