https://file.aiquickdraw.com/mathbot/file/1726934055711qaph9gvm.jpg

The problem in the image asks the following:

**Problem:**
Let \( H \) and \( K \) be two subgroups of a finite group \( G \). If \( H \subseteq K \), then show that \( [G : K] \) is a divisor of \( [G : H] \).

**Solution:**

We are given:
- \( G \) is a finite group.
- \( H \) and \( K \) are subgroups of \( G \).
- \( H \subseteq K \), meaning \( H \) is a subgroup of \( K \).
  
We need to show that \( [G : K] \) divides \( [G : H] \). Here, \( [G : H] \) denotes the index of \( H \) in \( G \), and similarly, \( [G : K] \) denotes the index of \( K \) in \( G \).

### Key Idea:
The index \( [G : H] \) is the number of left cosets of \( H \) in \( G \), i.e., the number of distinct sets of the form \( gH \) where \( g \in G \). Similarly, \( [G : K] \) is the number of left cosets of \( K \) in \( G \).

Since \( H \subseteq K \), we can consider the coset structure as follows:
1. First, consider the cosets of \( K \) in \( G \). There are \( [G : K] \) of these cosets.
2. Now, within each coset of \( K \), we can break it down into cosets of \( H \). Since \( H \subseteq K \), each coset of \( K \) contains some number of cosets of \( H \), specifically \( [K : H] \) cosets.

Thus, \( [G : H] = [G : K] \times [K : H] \).

This equation directly shows that \( [G : K] \) divides \( [G : H] \), since \( [G : H] \) is a product of \( [G : K] \) and \( [K : H] \).

### Conclusion:
We have shown that \( [G : K] \) divides \( [G : H] \) as required.

Would you like more details or have any questions about this? 

Here are some related questions to expand on the concept:
1. What is the meaning of the index \( [G : H] \) in group theory?
2. Can this result be extended to infinite groups?
3. How does the concept of normal subgroups affect the index \( [G : H] \)?
4. What is the relationship between the Lagrange's theorem and this problem?
5. How can this divisor relationship be visualized in terms of group actions?

**Tip:** Whenever you encounter cosets in a group, breaking them down step by step into smaller cosets (as done here with \( H \subseteq K \)) can often reveal divisor relationships in group indices.

Let H and K be two subgroups of a finite group G. If H ⊆ K, then show that [G:K] is a divisor of [G:H].

This solution demonstrates how, for subgroups H and K of a finite group G, with H being a subset of K, the index [G:K] is a divisor of [G:H]. This involves applying coset and subgroup index properties, supported by Lagrange's Theorem.

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