The question asks which statements are correct for a finite abelian group $G$ of order $n$. Let us analyze each statement:

(a) If $d$ divides $n$, then there exists a subgroup of $G$ of order $d$.

This is true for finite abelian groups. By the Fundamental Theorem of Finite Abelian Groups, any finite abelian group can be expressed as a direct product of cyclic groups. If $d$ divides $n$, there is a subgroup of $G$ of order $d$.

Correct.

(b) If $d$ divides $n$, then there exists an element of order $d$ in $G$.

This statement is also true for finite abelian groups. By Cauchy’s theorem and properties of abelian groups, if $d$ divides $n$, there must exist an element in $G$ whose order is $d$.

Correct.

(c) If every proper subgroup of $G$ is cyclic, then $G$ is cyclic.

This is true. If every proper subgroup of a finite abelian group $G$ is cyclic, $G$ itself must also be cyclic. This is a well-known property of finite abelian groups.

Correct.

(d) If $H$ is a subgroup of $G$, there exists a subgroup $N$ of $G$ such that $G/N \cong H$.

This statement is false. The existence of a subgroup $N$ such that $G/N \cong H$ does not hold in general for arbitrary subgroups $H$. This property requires $H$ to be a normal subgroup, but not every subgroup of an abelian group satisfies this.

Incorrect.

Final Answer:

Statements (a), (b), and (c) are correct. Statement (d) is incorrect.

Would you like a detailed explanation of any specific statement or concept?

Follow-up Questions:

1. What is the Fundamental Theorem of Finite Abelian Groups?
2. Can you explain Cauchy’s theorem and its relevance here?
3. Why is $G$ cyclic if every proper subgroup of $G$ is cyclic?
4. Can you give an example of an abelian group illustrating statements (a) and (b)?
5. Why does $G/N \cong H$ fail in (d) for general $H$?

Tip:

For any finite abelian group, knowing its prime factorization and direct product decomposition can simplify subgroup and element-order calculations.