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 $G$.)
$G=\langle\mathbf{R},+\rangle, H=\{\log n: n \in \mathbb{Z}, n>0\} . \quad H$ is $\square$ is not $\square$ a subgroup of $G$

Key Concepts

Subgroup Test

The subgroup test is a method used to determine whether a given subset of a group is a subgroup. Commonly, this test involves checking for non-emptiness, closure under the group operation, and closure under taking inverses. Alternatively, for groups with a single binary operation, it can be sufficient to show that for any two elements in the subset, the product of one element with the inverse of the other is still in the subset.

Closure Under Inverses

Closure under inverses is a key requirement in subgroup criteria. It mandates that for every element in the potential subgroup, its inverse (with respect to the group operation) must also be in the subset. This condition ensures that every element has an undoing element within the set, which is essential for the set to exhibit group behavior.

Group

A group is an algebraic structure consisting of a set equipped with an operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for every element. This concept forms the basis for much of abstract algebra and helps in understanding the structure and behavior of mathematical objects under a defined operation.

Subgroup

A subgroup is a subset of a group that is itself a group under the operation of the larger group. For a subset to qualify as a subgroup, it must include the identity element, be closed under the group operation, and include inverses for all of its elements. This concept emphasizes the idea that smaller groups can reside naturally within larger groups, sharing the same structural properties.

Transcript

Please give Ace some feedback