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The question in the image asks to prove the following statement:

**Problem**:
"Prove that if \( G \) is a finite group and \( a \in G \), then the order of \( a \) divides the order of \( G \)."

**Approach**:
This can be solved using *Lagrange's Theorem*.

### Proof Outline:
1. **Lagrange's Theorem**: 
   This theorem states that for any finite group \( G \), the order (i.e., the number of elements) of any subgroup \( H \) of \( G \) divides the order of \( G \).

2. **Consider the cyclic subgroup generated by \( a \)**: 
   Let \( \langle a \rangle \) denote the cyclic subgroup of \( G \) generated by \( a \). The order of this cyclic subgroup, \( |\langle a \rangle| \), is the same as the order of the element \( a \), which is denoted by \( o(a) \).

3. **Applying Lagrange's Theorem**:
   Since \( \langle a \rangle \) is a subgroup of \( G \), by Lagrange's theorem, the order of \( \langle a \rangle \) (i.e., \( o(a) \)) must divide the order of \( G \), denoted \( |G| \).

Thus, **the order of any element \( a \in G \) divides the order of the group \( G \)**.

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Would you like a more detailed explanation on any specific part? Or further clarifications on the theorem?

Here are some additional questions related to this topic:
1. What are some examples of groups where the order of elements divides the order of the group?
2. How does the order of an element help classify the structure of a group?
3. What happens if \( G \) is infinite? Does Lagrange's theorem still apply?
4. Can the order of an element in a group ever be larger than the order of the group?
5. How does this theorem relate to cosets in group theory?

**Tip**: Lagrange’s Theorem is crucial in understanding the structure of finite groups and plays a vital role in proofs involving divisibility and subgroup orders.

Buktikan jika G grup berhingga dan a ∈ G, maka order dari a membagi habis |G|.

Learn how to prove that if G is a finite group and a ∈ G, then the order of a divides the order of G using Lagrange's Theorem. This proof involves key group theory concepts and is suitable for undergraduate mathematics students.

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