Let $G$ be a finite abelian group of order $n$ . Then choose the correct statement.
a) If d divides n, then there exist a subgroup of $G$ of order $d$
b) If d divides n, then there exist an element of $G$ of order $d$
c) If every proper subgroup of $G$ is cyclic then $G$ is cyclic.
d) If $H$ is a subgroup of $G$, then there exist a subgroup $N$ such that $\frac{G}{N} \cong H$.
for a) I know that the converse of the Lagrange's Theorem for abelian group is true.
for b) and d) Take $G = \mathbb Z_2 \times \mathbb Z_2\times \mathbb Z_2 \times \mathbb Z_2 $ has no element of order 4.
for c) take $G= \mathbb Z_2 \times Z_2$ , every subgroup of $G$ is cyclic but $G$ is not cyclic .
I would be thankful if someone checks my solution.
