A mathematical group is a set of elements combined with an operation that satisfies four key properties: closure, associativity, identity, and invertibility. Closure means that performing the operation on any two elements in the group results in another element within the same group. Associativity ensures that the order in which operations are performed does not affect the outcome. The identity element is a special element in the group that, when combined with any other element, leaves it unchanged. Invertibility means that for every element in the group, there exists another element that can combine with it to produce the identity element. Groups are fundamental in various areas of mathematics, including algebra and geometry.