In each of the following, determine whether or not H is a...

In each of the following, determine whether or not $H$ is a subgroup of $G$. (Assume that the operation of $H$ is the same as that of $G$.)
$G=\langle R,+\rangle, H=\{\log a: a \in \mathbb{Q}, a>0\} . \quad H$ is $\square \quad$ is not $\square$ a subgroup of $G$

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Key Concepts

Group

A group is an algebraic structure consisting of a set together with an operation that combines any two elements to form a third element. The operation must satisfy four basic properties: closure, associativity, the existence of an identity element, and the existence of inverse elements for every element in the set.

Subgroup

A subgroup is a subset of a group that itself forms a group under the same operation. For a subset to be considered a subgroup, it must contain the identity element of the larger group, be closed under the group operation, and include the inverses of all its elements.

Subgroup Test

The subgroup test provides a streamlined method for determining whether a subset of a group is also a group. It typically requires proving that the subset is nonempty and that for any two elements in the subset, the operation combining one element with the inverse of the other results in an element that remains within the subset.

Homomorphism

A homomorphism is a structure-preserving map between two groups that respects the group operations. Functions like the logarithm can serve as homomorphisms by converting multiplicative relationships into additive ones, thus linking different group structures and allowing properties to be studied in an alternative context.

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