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To prove **Lagrange's Theorem**, let's proceed with a formal approach.

### Lagrange's Theorem Statement:
Let \( G \) be a finite group, and \( H \) a subgroup of \( G \). Then the order of \( H \) (i.e., \( |H| \)) divides the order of \( G \) (i.e., \( |G| \)).

### Key Concepts Needed:
The proof relies on the concept of **cosets** of \( H \) in \( G \):
1. A **left coset** of \( H \) with respect to an element \( g \in G \) is defined as \( gH = \{ gh : h \in H \} \).
2. The group \( G \) can be partitioned into disjoint left cosets of \( H \).
3. Each coset \( gH \) has the same number of elements as \( H \).

### Properties of Cosets (given in the problem):
1. **If \( t \in sH \), then \( sH = tH \):** This states that if two cosets intersect (i.e., share at least one element), they are actually identical.

2. **If \( t \notin sH \), then \( sH \cap tH = \emptyset \):** If two cosets are not the same, they do not share any elements (disjoint).

3. **\( |sH| = |H| \):** Each left coset \( sH \) of \( H \) in \( G \) has the same number of elements as \( H \).

### Proof of Lagrange's Theorem:
1. **Partition of \( G \):** 
   Since \( G \) is finite, we can list its elements. The set \( G \) can be divided into disjoint left cosets of \( H \) (by properties 1 and 2 above). That is, each element of \( G \) belongs to exactly one coset of \( H \).

2. **Equal Size of Cosets:**
   By property 3, each coset \( sH \) has the same number of elements as \( H \). Therefore, every coset has \( |H| \) elements.

3. **Counting Elements in \( G \):**
   Let the number of distinct left cosets of \( H \) in \( G \) be \( k \). Then, the total number of elements in \( G \) can be counted by counting the elements in each coset, giving
   \[
   |G| = k \cdot |H|.
   \]

4. **Divisibility Conclusion:**
   Since \( |G| = k \cdot |H| \), \( |H| \) divides \( |G| \), proving Lagrange's Theorem.

### Summary:
This proves that for any finite group \( G \) and any subgroup \( H \) of \( G \), the order of \( H \) divides the order of \( G \).

Would you like further clarification on any part of this proof? 

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Here are some related questions that expand on the topic:

1. How would Lagrange's Theorem apply to infinite groups, if at all?
2. Can Lagrange's Theorem be used to determine all possible orders of subgroups of a group \( G \) with a known order?
3. How does Lagrange’s Theorem help in proving that groups of prime order are cyclic?
4. What are the implications of Lagrange’s Theorem for the structure of a group with prime power order?
5. How would the proof change if we used right cosets instead of left cosets?

**Tip:** When working with cosets, remember that they form either a partition or cover of the group, ensuring every element in \( G \) is part of exactly one coset. This idea underpins many group theory proofs.

Prove Lagrange's Theorem for a finite group G and a subgroup H, showing that |H| divides |G|.

Explore the proof of Lagrange's Theorem in group theory, which shows that for a finite group G and a subgroup H, the order of H divides the order of G. This proof involves the concept of cosets and properties related to finite groups.

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