H` is a subgroup of G/H iff H`=K/H where K is subgroup of G containing H. 

The statement can be interpreted and formalized in the context of group theory and quotient groups. Let's break it down:

- Let \( G \) be a group and \( H \) a normal subgroup of \( G \).
- \( H' \) is a subgroup of \( G/H \) if and only if there exists a subgroup \( K \) of \( G \) containing \( H \) such that \( H' = K/H \).

### Explanation:

1. **Subgroups of Quotient Groups**: 
   For any normal subgroup \( H \) of \( G \), the quotient group \( G/H \) consists of the cosets of \( H \) in \( G \). A subgroup \( H' \subseteq G/H \) corresponds to some set of cosets. The statement is asserting that any such subgroup \( H' \) arises from a subgroup \( K \subseteq G \) that contains \( H \).

2. **Relation Between Subgroups**:
   The subgroup \( H' \) of \( G/H \) can be written as \( H' = K/H \), where \( K \) is a subgroup of \( G \) containing \( H \). The elements of \( H' \) are cosets of \( H \), and each coset \( kH \) (where \( k \in K \)) corresponds to an element in the quotient group.

3. **Subgroup Condition**:
   The requirement that \( K \) contains \( H \) ensures that the quotient \( K/H \) makes sense, and this quotient becomes a subgroup of \( G/H \).

### Why This Holds:
The subgroups of a quotient group \( G/H \) are in one-to-one correspondence with the subgroups \( K \) of \( G \) that contain \( H \). Specifically, there is a **lattice isomorphism** between the subgroups of \( G \) that contain \( H \) and the subgroups of \( G/H \). This is known as the **correspondence theorem** or the **fourth isomorphism theorem** in group theory.

### Summary of Theorem:

- If \( H' \subseteq G/H \) is a subgroup of \( G/H \), then there exists a subgroup \( K \subseteq G \), with \( H \subseteq K \subseteq G \), such that \( H' = K/H \).
  
- Conversely, for every subgroup \( K \) of \( G \) that contains \( H \), the quotient \( K/H \) is a subgroup of \( G/H \).

Let me know if you'd like further clarification or details!

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Here are some related questions:

1. What are the conditions for a subgroup \( H \) to be normal in \( G \)?
2. Can you explain the correspondence theorem in more detail?
3. How do cosets relate to quotient groups?
4. What is the significance of normal subgroups in group theory?
5. How can we construct quotient groups in specific examples?

**Tip**: When working with quotient groups, always verify that the subgroup used in the quotient is normal, as this is essential for constructing the group correctly.

H` is a subgroup of G/H iff H`=K/H where K is subgroup of G containing H.

Explore the relationship between a subgroup H' of the quotient group G/H and its corresponding subgroup K in G containing H, based on the Correspondence Theorem in group theory. Understand the structure of subgroups in quotient groups and how cosets function.

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