h subgroup g then it implies the product of h and inverse of h = H

If \( H \) is a subgroup of a group \( G \), then for every element \( h \in H \), the product of \( h \) and its inverse \( h^{-1} \) must satisfy certain properties. Specifically, in group theory:

1. **Identity Element**: For any \( h \in H \), \( h \cdot h^{-1} = e \), where \( e \) is the identity element of \( G \). This means multiplying any element of \( H \) by its inverse returns the identity element of the group.
   
2. **Closure**: Since \( H \) is a subgroup, it is closed under group operations. Therefore, the inverse \( h^{-1} \in H \). This means \( h^{-1} \) is also an element of \( H \).

Thus, when you say that the product of \( h \) and its inverse is in \( H \), it is really stating that for any \( h \in H \), \( h \cdot h^{-1} = e \in H \). Since \( e \) is the identity, it naturally belongs to any subgroup.

To summarize:
- The product of any element \( h \) and its inverse \( h^{-1} \) always equals the identity element \( e \), which is in \( H \).
- If \( H \) is a subgroup, both \( h \) and \( h^{-1} \) are in \( H \), and their product gives the identity, which is a required property for \( H \) being a subgroup.

Would you like more details on any part of this explanation?

### Related Questions:
1. What are the criteria for a subset to be considered a subgroup?
2. How do we prove that the inverse of an element in a subgroup is also in the subgroup?
3. What is the identity element in group theory?
4. Can there be multiple identity elements in a group?
5. What is the difference between a subgroup and a coset?

### Tip: 
In group theory, always ensure that the identity element and the inverse of each element exist and belong to the set to verify it as a subgroup.

Explore the concept of subgroups in group theory and learn why the product of an element and its inverse equals the identity element. This detailed explanation covers important mathematical concepts such as the identity element and the closure property of subgroups. Suitable for undergraduate students studying abstract algebra.

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