The text in the image appears to be a mathematical problem written in Indonesian, discussing group theory. It mentions groups $G$, subgroups $H$ and $N$, and normal subgroups, asking to prove that $HN$ is a normal subgroup of $G/N$ if and only if $H$ is a normal subgroup of $G$.

Here is a structured explanation:

Problem Analysis

1. Given:

   • $G$: A group.
   • $H$: A subgroup of $G$.
   • $N$: A normal subgroup of $G$, where $N \leq H$ (i.e., $N$ is a subgroup of $H$).
   • It is stated that $HN$ is the set of all elements that can be expressed as $h \cdot n$ where $h \in H$ and $n \in N$.

2. To Prove:

   • $HN$ is a normal subgroup of $G/N$ if and only if $H$ is a normal subgroup of $G$.

Approach to Proof

• To show $HN \triangleleft G/N$, you need to verify the normal subgroup condition in the quotient group $G/N$.
• The equivalence condition requires:
  1. $H \triangleleft G$ (normal subgroup).
  2. $N \leq H$, which ensures the interaction between $H$, $N$, and $G/N$.

Would you like a full step-by-step proof or additional insights into group theory concepts? Here are some related questions for further exploration:

Related Questions:

1. What does it mean for a subgroup to be normal, and how can it be shown using group elements?
2. How is the operation in a quotient group $G/N$ defined?
3. Why is the condition $N \leq H$ important for the result?
4. What is the structure of $HN$, and how does it interact with the group $G/N$?
5. Can you generalize this result to other mathematical structures (e.g., rings or modules)?

Tip: Visualizing group and subgroup relationships using set diagrams or Cayley diagrams can simplify these abstract concepts. Let me know if you'd like a diagram!