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

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

Translation: "Prove that if $G$ is a finite group and $a \in G$, then the order of $a$ divides the order of $G$."

This is essentially a direct application of Lagrange's Theorem in group theory. Here's how to approach the proof:

Proof Outline:

1. Group Definition and Order:

   • Let $G$ be a finite group, so the order of the group is $|G|$ (the number of elements in $G$).
   • Let $a \in G$. The order of the element $a$, denoted by $o(a)$, is the smallest positive integer $m$ such that $a^m = e$, where $e$ is the identity element of $G$.

2. Subgroup Generated by $a$:

   • The element $a$ generates a cyclic subgroup $\langle a \rangle$ of $G$, consisting of all powers of $a$, i.e., $\langle a \rangle = \{e, a, a^2, \dots, a^{o(a)-1}\}$.
   • The order of this subgroup is exactly $o(a)$, by the definition of the order of an element.

3. Lagrange's Theorem:

   • Lagrange's Theorem states that for any finite group $G$, the order of any subgroup $H$ of $G$ divides the order of $G$, i.e., $|H|$ divides $|G|$.
   • In our case, the subgroup $\langle a \rangle$ has order $o(a)$, so by Lagrange's theorem, $o(a)$ divides $|G|$.

Thus, we've shown that the order of the element $a$ divides the order of the group $G$.

Would you like further details or clarifications on this proof?

Here are 5 related questions for deeper exploration:

1. What is the significance of cyclic subgroups in group theory?
2. How does Lagrange's Theorem apply to infinite groups?
3. Can the order of an element ever be larger than the order of the group?
4. What are some examples of groups where Lagrange's Theorem is particularly useful?
5. How does Lagrange's Theorem relate to cosets?

Tip: When studying group theory, always consider whether an element generates a subgroup and how properties like the order of elements and subgroups help in understanding the structure of the group.