We are asked to prove that $G_n = \{ a \in G \mid a = x^n \text{ for some } x \in G \}$ is a subgroup of the group $G$, which is abelian.

Steps to prove that $G_n$ is a subgroup of $G$:

To prove that $G_n$ is a subgroup of $G$, we need to show that:

1. The identity element of $G$ belongs to $G_n$.
2. $G_n$ is closed under the group operation.
3. $G_n$ is closed under taking inverses.

Step 1: Identity element in $G_n$

• Let $e$ be the identity element of $G$.
• Since $G$ is abelian, for any $x \in G$, we have $e = x^0$ for $n = 0$.
• Therefore, $e \in G_n$, so the identity element is in $G_n$.

Step 2: Closure under the group operation

• Let $a_1, a_2 \in G_n$, so by definition, there exist $x_1, x_2 \in G$ such that $a_1 = x_1^n$ and $a_2 = x_2^n$.
• We need to show that $a_1 a_2 \in G_n$. Since $G$ is abelian, we have: $a_1 a_2 = x_1^n x_2^n = (x_1 x_2)^n.$
• Because $x_1 x_2 \in G$ (since $G$ is a group and closed under the group operation), we see that $a_1 a_2 \in G_n$.

Step 3: Closure under inverses

• Let $a \in G_n$. By definition, there exists $x \in G$ such that $a = x^n$.
• We need to show that $a^{-1} \in G_n$. Since $G$ is abelian and the inverse of an element in a group is also in the group, we have: $a^{-1} = (x^n)^{-1} = (x^{-1})^n.$
• Since $x^{-1} \in G$, it follows that $a^{-1} \in G_n$.

Conclusion:

Since $G_n$ contains the identity element, is closed under the group operation, and is closed under inverses, it satisfies the three conditions of being a subgroup. Therefore, $G_n$ is a subgroup of $G$.

Would you like more details on any of these steps?

Here are 5 related questions:

1. What is the definition of a subgroup in group theory?
2. Why is the closure under inverses important in proving a subgroup?
3. How does the property of $G$ being abelian simplify this proof?
4. Can we generalize this proof to non-abelian groups?
5. What would change in this proof if $G_n$ was not defined with powers of $n$?

Tip: When proving a subgroup, it's useful to always verify identity, closure under the operation, and closure under inverses—these are the core criteria.