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?

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.