for a h to be subgroup of g  the  required condition if a,b belongs to h

For a subset \( H \) to be a subgroup of a group \( G \), the following conditions must be satisfied for all elements \( a, b \in H \):

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

These are the three fundamental conditions required for \( H \) to be a subgroup of \( G \).

Would you like further details or examples about subgroup conditions? 

Here are some related questions:
1. How do you prove a given set is a subgroup of a group?
2. Can a subgroup be the same as the group itself?
3. What are examples of non-trivial subgroups in common groups like \( \mathbb{Z} \) or \( \mathbb{R} \)?
4. What is the difference between normal subgroups and regular subgroups?
5. How does Lagrange's Theorem relate to subgroups?

**Tip**: To check if a subset is a subgroup, start by verifying closure under the group operation and checking for the identity and inverse elements.

For a h to be subgroup of g the required condition if a,b belongs to h.

Explore the key conditions for a subset H to be a subgroup of a group G in group theory. This solution covers the subgroup criterion, including closure, identity, and inverses. Suitable for students studying undergraduate-level abstract algebra.

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