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To solve question 1:

### Problem:
Let \( G \) be a group, \( H \) and \( N \) be subgroups of \( G \). If \( N \) is a normal subgroup of \( G \), prove that \( H \cap N \) is a normal subgroup of \( H \).

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### Solution:

1. **Normal subgroup definition**: 
   \( N \) is a normal subgroup of \( G \), so for any \( g \in G \) and \( n \in N \), we have:
   \[
   g n g^{-1} \in N.
   \]

2. **Intersection \( H \cap N \)**:
   The subgroup \( H \cap N \) consists of all elements that are in both \( H \) and \( N \):
   \[
   H \cap N = \{x \in G : x \in H \text{ and } x \in N\}.
   \]

3. **Goal**:
   Prove that \( H \cap N \) is a normal subgroup of \( H \). For this, we need to show:
   \[
   \forall h \in H, \forall x \in H \cap N, \; h x h^{-1} \in H \cap N.
   \]

4. **Verification**:
   Let \( h \in H \) and \( x \in H \cap N \). Since \( x \in H \cap N \), we have:
   - \( x \in H \),
   - \( x \in N \).

   Since \( h \in H \) and \( N \) is normal in \( G \), we know:
   \[
   h x h^{-1} \in N.
   \]

   Additionally, since \( h \in H \) and \( x \in H \), the group \( H \) being closed under conjugation implies:
   \[
   h x h^{-1} \in H.
   \]

   Combining these, we have:
   \[
   h x h^{-1} \in H \cap N.
   \]

5. **Conclusion**:
   Since \( h x h^{-1} \in H \cap N \) for all \( h \in H \) and \( x \in H \cap N \), \( H \cap N \) is a normal subgroup of \( H \).

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If you have further questions, feel free to ask!

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### Related Questions:

1. Can you provide examples of groups \( G \), \( H \), and \( N \) satisfying the problem's conditions?
2. How can we extend this result if \( H \) is not a subgroup of \( G \)?
3. Why is the normality of \( N \) in \( G \) critical for this proof?
4. What is the significance of normal subgroups in group theory?
5. How does this result generalize to intersections of multiple normal subgroups?

### Tip:
Always verify the subgroup properties (closure, associativity, identity, and inverses) before proving normality.

Let G be a group, H and N be subgroups of G. If N is a normal subgroup of G, prove that H ∩ N is a normal subgroup of H.

Learn how to prove that the intersection of a normal subgroup N of G with a subgroup H is a normal subgroup of H. This solution explains key concepts in group theory and normal subgroups for undergraduate abstract algebra.

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