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.