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buktikan dengan cara sistematis menggunakan Teorema tersebut 

The question asks to prove that if \( G \) is a finite group with order \( n \), then for every element \( a \in G \), it holds that \( a^n = e \), where \( e \) is the identity element of the group. The hint suggests using Lagrange's Theorem to establish this result.

### **Proof using Lagrange's Theorem:**

#### **Step 1: Subgroup generated by \( a \)**
For any element \( a \in G \), the set of powers of \( a \), denoted by:
\[
\langle a \rangle = \{a^k \mid k \in \mathbb{Z}\}
\]
is a cyclic subgroup of \( G \) generated by \( a \). The order of this subgroup, denoted as \( |\langle a \rangle| \), is the smallest positive integer \( m \) such that \( a^m = e \). This \( m \) is called the *order* of the element \( a \).

#### **Step 2: Application of Lagrange's Theorem**
Lagrange's Theorem states that the order of any subgroup of a finite group divides the order of the group. Since \( \langle a \rangle \) is a subgroup of \( G \), the order of \( \langle a \rangle \) (denoted \( m \)) must divide \( n \), the order of \( G \). Hence, there exists some integer \( k \) such that:
\[
n = km
\]
where \( m \) is the order of \( a \) (i.e., the smallest integer such that \( a^m = e \)).

#### **Step 3: Conclusion**
Since \( a^m = e \) by the definition of the order of \( a \), and \( m \) divides \( n \), we can write:
\[
a^n = a^{km} = (a^m)^k = e^k = e
\]
Thus, \( a^n = e \) for every \( a \in G \).

### Conclusion:
We have proved that for any finite group \( G \) with order \( n \), for every element \( a \in G \), \( a^n = e \), using Lagrange’s Theorem.

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Would you like further clarification or details on any part of the proof? Here are some related questions you might explore:

1. What are the conditions for Lagrange’s Theorem to hold?
2. How can this result be applied to cyclic groups specifically?
3. What is an example of a group where this theorem applies?
4. How does this proof differ if \( G \) is infinite?
5. What is the significance of the identity element \( e \) in group theory?

**Tip:** Lagrange's Theorem is extremely useful in determining the possible orders of elements in a group, especially when working with finite groups.

Buktikan jika G grup berhingga dengan order n, maka a^n = e untuk setiap a ∈ G.

Learn how to prove that for any finite group G of order n, a^n = e for all a in G. This proof utilizes Lagrange's Theorem and covers key concepts in group theory, suitable for undergraduate abstract algebra students.

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