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.