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To prove that \( G \) is a finite group, we need to confirm that it satisfies the four properties of a group: closure, associativity, identity, and inverses. Given the problem statement, let’s go through each property:

1. **Closure**: \( G \) is closed under the multiplication operation by assumption.

2. **Associativity**: The operation on \( G \) is associative by assumption.

3. **Identity**: The element \( 1 \in G \) is given as an identity element such that \( 1 \cdot g = g \cdot 1 = g \) for all \( g \in G \).

4. **Inverses**: We need to show that every element in \( G \) has an inverse. 

### Proof of Inverses
Since \( G \) is finite and the cancellation law holds (i.e., if \( g \cdot h = g \cdot k \), then \( h = k \)), each element \( g \in G \) must act bijectively under multiplication. Therefore, each element has a unique "inverse" that brings it back to the identity.

#### Steps:
1. For any \( g \in G \), consider the set \( \{ g \cdot x : x \in G \} \).
2. Since \( G \) is finite and \( g \) satisfies the cancellation property, the map \( x \mapsto g \cdot x \) is injective.
3. In a finite set, an injective map is also surjective, so \( g \cdot x = 1 \) for some \( x \in G \).
4. This element \( x \) is the inverse of \( g \), satisfying \( g \cdot x = x \cdot g = 1 \).

Since \( G \) satisfies all group properties, we conclude that \( G \) is a finite group.

Would you like further details on this proof or have questions about any specific step?

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Here are some follow-up questions related to this proof:

1. How does the finiteness of \( G \) help in proving the existence of inverses?
2. Why is the cancellation law important in proving that \( G \) is a group?
3. How does the injective nature of the map \( x \mapsto g \cdot x \) imply surjectivity in a finite set?
4. What is the significance of an identity element in group theory?
5. Could this proof be extended to an infinite set \( G \) under the same assumptions?

**Tip:** When dealing with group proofs, verifying the fundamental properties (closure, associativity, identity, and inverses) is key to confirming the group structure.

Let G be a finite set which is closed under an associative multiplication operation. Suppose that there exists 1 ∈ G with 1 ⋅ g = g = g ⋅ 1 and the Cancellation Law holds, i.e., if g, h, k ∈ G with g ⋅ h = g ⋅ k, then h = k. Prove that G is a finite group.

Explore the proof that a finite set with an associative operation, identity, and cancellation law forms a group. This problem involves fundamental group theory concepts and Lagrange's Theorem, suitable for abstract algebra studies.

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