We are tasked with proving that if $H$ is a subgroup of $G$ and the index $|G:H| = 2$, then $H$ is a normal subgroup of $G$.

Step-by-Step Proof:

1. Understanding the index $|G:H| = 2$:

   • The index of $H$ in $G$, denoted $|G:H|$, is the number of distinct cosets of $H$ in $G$. If $|G:H| = 2$, this means that there are exactly two cosets of $H$ in $G$.
   • These two cosets are: $H \quad \text{and} \quad gH \quad \text{for some} \quad g \in G \setminus H.$

2. Definition of normality:

   • A subgroup $H$ is normal in $G$ (denoted $H \triangleleft G$) if for every $g \in G$, the left and right cosets of $H$ are the same. This means that for all $g \in G$, we have: $gH = Hg.$

3. Cosets structure:

   • Since $|G:H| = 2$, the two cosets of $H$ are $H$ itself and $gH$, where $g \in G \setminus H$.
   • So, $G$ can be written as the disjoint union of two cosets: $G = H \cup gH.$

4. Key observation:

   • Let $g \in G \setminus H$. Since $G$ has only two cosets, for any $g \in G \setminus H$, the coset $gH$ must be equal to $Hg$. This is because the set of right cosets of $H$ in $G$ also has only two elements: $H$ and $Hg$.
   • If $gH \neq Hg$, then we would have more than two cosets, which contradicts the assumption that $|G:H| = 2$.

5. Conclusion:

   • Since $gH = Hg$ for all $g \in G$, it follows that $H$ is normal in $G$. Hence, $H \triangleleft G$.

Thus, we have shown that if $|G:H| = 2$, then $H$ is a normal subgroup of $G$.

---

Do you want more details or have any further questions?

Here are 5 related questions to expand on this concept:

1. What are examples of groups $G$ and subgroups $H$ where $|G:H| = 2$?
2. How does the result change if $|G:H| = 3$ instead of 2?
3. What is the definition of cosets, and how are they used to measure the index of a subgroup?
4. Can you explain the connection between normal subgroups and quotient groups?
5. What other methods can be used to determine if a subgroup is normal?

Tip: If the index of a subgroup is 2, it is automatically normal because the group structure forces the left and right cosets to coincide.