Final Answer:
The union of two subgroups of a group need not necessarily be a subgroup. Counterexamples can be found in specific groups and subgroups that violate closure under the group operation, which is a fundamental requirement for a set to be a subgroup.
Step-by-step explanation:
To demonstrate that the union of two subgroups may not be a subgroup, consider the group Z_4, the integers modulo 4, under addition. Let H = {0, 2} and K = {0, 1, 3} be two subgroups of Z_4. The union of H and K is {0, 1, 2, 3}, which is not a subgroup.
To establish this, we can observe that 1 + 3 = 0 is not in the union {0, 1, 2, 3}. The closure property is violated, as the result of the group operation is outside the set. Therefore, the union of H and K does not satisfy the conditions required to be a subgroup.
This counterexample illustrates that the union of subgroups might not form a subgroup due to the failure of closure under the group operation. It emphasizes the importance of all subgroup criteria, including closure, associativity, and the existence of inverses, to ensure that a subset is a valid subgroup of a given group.