Subgroup Criterion Using Inverses

Suppose we have a non-empty subset of a group $G$. Prove that this subset is a subgroup if and only if for all $x, y \in H$, $xy^{-1} \in H$.

In the realm of group theory, determining whether a subset of a group forms a subgroup can be a central question. This problem utilizes a significant criterion involving the inverses of elements. The problem at hand asks you to show that a subset of a group is a subgroup if and only if for all its elements x and y, the element obtained by taking the product of x and the inverse of y is also within the subset. This criterion streamlines the verification of subgroup properties.

Understanding this problem relies heavily on grasping the definition of a subgroup, which must satisfy closure under the group operation, contain the identity element, and every element must have an inverse within the set. The given criterion offers a compact way to validate these properties implicitly. Recognizing that if xy inverse is in the subset for all elements x and y, closure and the existence of inverses are essentially wrapped into one condition.

This problem also illustrates the abstract nature of group theory, where properties such as closure and inverses are explored through logical reasoning rather than explicit enumeration of elements. The use of inverses is particularly noteworthy as it ties into themes of symmetry and structure within algebraic systems, offering a glimpse into the elegance and power of algebraic abstractions.