To show that $H$ is a subgroup of $G$, we will use the definition of $H$ and apply the subgroup criteria. The set $H$ is defined as the intersection of all subgroups of $G$ that contain $X$. We need to demonstrate that $H$ satisfies the subgroup properties.

Step-by-Step Solution

1. Definition of $H$:

Let $H = \bigcap_{\alpha} H_\alpha$, where each $H_\alpha$ is a subgroup of $G$ that contains $X$. In other words, $H$ consists of all elements of $G$ that are in every subgroup $H_\alpha$ containing $X$.

2. Subgroup Criterion:

To show that $H$ is a subgroup of $G$, we need to verify that it satisfies the following three conditions:

1. $H$ is non-empty.
2. $H$ is closed under the group operation.
3. $H$ is closed under taking inverses.

3. Non-emptiness:

Since $G$ is a group, it contains the identity element $e$. Every subgroup of $G$ must contain $e$, because the identity is a required element of any group. Hence, each $H_\alpha$ contains $e$, so $e \in H$. Therefore, $H$ is non-empty because it contains at least the identity element $e$.

4. Closed under the group operation:

Let $a, b \in H$. By the definition of $H$, this means that $a, b \in H_\alpha$ for every subgroup $H_\alpha$ containing $X$. Since each $H_\alpha$ is a subgroup, and subgroups are closed under the group operation, $ab \in H_\alpha$ for every $H_\alpha$. Therefore, $ab \in H$. Thus, $H$ is closed under the group operation.

5. Closed under taking inverses:

Let $a \in H$. By the definition of $H$, $a \in H_\alpha$ for every subgroup $H_\alpha$ containing $X$. Since each $H_\alpha$ is a subgroup, and subgroups are closed under taking inverses, $a^{-1} \in H_\alpha$ for every $H_\alpha$. Therefore, $a^{-1} \in H$. Thus, $H$ is closed under taking inverses.

Conclusion:

Since $H$ is non-empty, closed under the group operation, and closed under taking inverses, $H$ satisfies the subgroup criteria. Therefore, $H$ is a subgroup of $G$.

This subgroup $H$ is called the subgroup of $G$ generated by the subset $X$ because it is the smallest subgroup of $G$ containing $X$.

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Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How does the subgroup generated by $X$ relate to the concept of a normal subgroup?
2. Can the subgroup generated by $X$ be equal to $G$? Under what conditions?
3. If $X$ is already a subgroup, how does $H$ change?
4. How can you describe the subgroup generated by multiple elements?
5. What is the free group generated by a set, and how does it relate to this subgroup?

Tip: Remember that any subgroup must always contain the identity element, inverses, and be closed under the group operation. This is crucial when determining subgroups.