Math question
let H be a subset of Group G such that if a, b are elements of H, then (a^(-1)) ((b^(-1)) is also an element of H
Does it follow that H is a subgroup of G?

To determine if \( H \) is a subgroup of \( G \), we need to check the subgroup criteria, which are:

1. **Closure**: If \( a \) and \( b \) are elements of \( H \), then \( ab \) must also be an element of \( H \).
2. **Identity element**: The identity element \( e \) of the group \( G \) must be in \( H \).
3. **Inverses**: For every element \( a \in H \), its inverse \( a^{-1} \) must also be in \( H \).

The condition given is: 
- If \( a \) and \( b \) are in \( H \), then \( (a^{-1})(b^{-1}) \) is also in \( H \).

This condition suggests that \( H \) might have the necessary properties for closure under the operation and inverses, but we still need to examine whether it guarantees the subgroup criteria.

### 1. **Closure**:
The condition provided, \( (a^{-1})(b^{-1}) \in H \) for \( a, b \in H \), does not directly tell us whether \( ab \in H \), which is required for closure. Therefore, the closure property is **not guaranteed** by this condition.

### 2. **Identity element**:
To check if the identity element \( e \) is in \( H \), consider that for any \( a \in H \), \( a^{-1} \) must also be in \( H \) (since \( (a^{-1})(a^{-1}) \in H \) by the condition). This suggests that the identity element might be in \( H \), because \( e^{-1} = e \), so we can expect \( e \) to be in \( H \) by virtue of the inverse condition.

### 3. **Inverses**:
If \( a \in H \), the condition ensures that \( a^{-1} \in H \), since \( a^{-1} \) will appear as part of the product \( (a^{-1})(a^{-1}) \), which is in \( H \).

### Conclusion:
Although the given condition ensures that inverses are in \( H \) and possibly the identity element is in \( H \), it does **not guarantee closure** (i.e., \( ab \in H \) for \( a, b \in H \)). Thus, **\( H \) is not necessarily a subgroup** of \( G \) based on this condition alone.

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Would you like more details or further exploration into any part of this? Here are some related questions to expand your understanding:

1. What does the closure property of a subgroup specifically require in terms of elements of the group?
2. How does the identity element relate to the inverse of an element in a subgroup?
3. What is the significance of the given condition \( (a^{-1})(b^{-1}) \in H \) in group theory?
4. Can you construct an example where the given condition holds but \( H \) is not a subgroup?
5. How would you prove closure in a subgroup more directly?

**Tip**: Always ensure the three subgroup conditions (closure, identity, and inverses) are verified when checking if a subset is a subgroup!

Let H be a subset of Group G such that if a, b are elements of H, then (a^(-1)) ((b^(-1)) is also an element of H. Does it follow that H is a subgroup of G?

Explore the conditions under which a subset of a group is a subgroup, based on the provided inverse property. Learn about closure, identity elements, and inverses in group theory. This detailed solution helps clarify subgroup criteria for students in undergraduate-level mathematics.

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