This is a question about group theory. It states that $H$ is a subgroup of $G$, and $e_1$ and $e_2$ are the identity elements for $H$ and $G$, respectively. The question asks for the relationship between $e_1$ and $e_2$.

Since $H$ is a subgroup of $G$, the identity element of $H$ is also the identity element of $G$. Therefore, $e_1 = e_2$.

The correct answer is: (A) $e_1 = e_2$.

Would you like further clarification or additional examples on this topic?

Further questions:

1. What is a subgroup, and how does it relate to a group?
2. Can there be different identity elements in subgroups of the same group?
3. What is the significance of the identity element in group theory?
4. How do we prove that a set is a subgroup of a group?
5. Can a group have multiple subgroups, and what are some examples?

Tip: In group theory, the identity element is central because it leaves other elements unchanged when combined with them.