Verify Subgroup Closure

Given a finite subset of a group, verify if the subset is closed under the group operation to determine if it is a subgroup.

Determining whether a finite subset of a group is a subgroup involves verifying that this subset is closed under the group operation. This concept is rooted in group theory, which is an abstract branch of mathematics dealing with algebraic structures known as groups. Groups and subgroups form a foundational element of abstract algebra, and understanding them is essential in studying the behavior of symmetrical objects and structures.

To check closure under a group operation, you should verify that performing the operation on any two elements of the subset yields another element that is still within the subset. This requires an understanding of the group's operation itself and how these operations apply to each element in the subset. Importantly, the subset must include the identity element of the group, and each element must have an inverse that is also within the subset.

This problem challenges your understanding of key concepts in group theory, such as the identity element and inverse elements, as well as your ability to systematically verify operations. It is a typical problem in the study of groups and subgroups that helps hone logical reasoning and abstraction skills, particularly for students dealing with finite groups of small order.