Let G be an abelian group. Let H={x ∈G: x=y^2. for some .y ∈G}, that is, let H be the set of all

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Let G be an abelian group. Let H={x ∈G: x=y^2. for some .y...

Let $G$ be an abelian group.
Let $H=\left\{x \in G: x=y^{2}\right.$ for some $\left.y \in G\right\}$, that is, let $H$ be the set of all the elements of $G$ which have a square root. Prove that $H$ is a subgroup of $G$.

Key Concepts

Inverse Elements

For every element in a group, there exists an inverse element such that when the element and its inverse are combined using the group operation, the result is the identity element. The existence of inverses for all elements in a subset is a necessary condition for the subset to qualify as a subgroup.

Identity Element

The identity element in a group is a unique element that, when combined with any element of the group using the group operation, leaves that element unchanged. For a subset to be a subgroup, it must include this identity element, ensuring the subset has a neutral starting point for its group operation.

Square Elements in a Group

Square elements in a group are those elements that can be expressed as the square of another element, meaning an element x is a square if there exists some y in the group such that x = y * y. In group theory, considering elements with square roots often helps in understanding the structure and properties of the group, as well as in constructing specific subgroups.

Subgroup

A subgroup is a subset of a group that is itself a group under the same operation. To be considered a subgroup, the subset must include the identity element, be closed under the group operation, and contain the inverse of each of its elements. Verifying these properties ensures that the structure of the group is maintained within the subset.

Abelian Group

An abelian group is a group in which the binary operation is commutative. This means that for any two elements a and b in the group, the equation a * b = b * a holds. This property simplifies many computations and proofs, as it allows one to rearrange terms without affecting the result.

Closure Under the Group Operation

Closure is the property that states that for any two elements in a set, the result of applying the group operation to these two elements is also in the set. In the context of proving a subset is a subgroup, showing closure ensures that combining any two elements of the subset via the group operation yields another element that is still in the subset.

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