If H is subgroup of G with |G:H| =2 then show that H is normal group of G

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 \( G \).

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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.

Learn how to prove that a subgroup H of a group G is normal if the index |G:H| = 2. This proof covers key group theory concepts such as cosets, the index of a subgroup, and normal subgroups. Ideal for undergraduate students studying abstract algebra.

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