The correct answer is:
d. None of the above
Let's go through the options one by one:
a. H ∩ K = {e}, the identity element: This is not necessarily true. The intersection of subgroups H and K might contain more elements than just the identity element. For example, consider the group G = Z6 (integers modulo 6) with subgroups H = {0, 2, 4} and K = {0, 3}. The intersection of H and K is {0}, which is the identity element, but this is not always the case.
b. H ∪ K = G: This is not necessarily true either. The union of subgroups H and K might not cover all elements of the group G. For example, consider the group G = Z8 (integers modulo 8) with subgroups H = {0, 4} and K = {0, 2, 4, 6}. The union of H and K is {0, 2, 4, 6}, which is not equal to G.
c. H = K: This is also not necessarily true. The subgroups H and K might be different even if their sizes are the same (|H| = |K|). For example, consider the group G = Z9 (integers modulo 9) with subgroups H = {0, 3, 6} and K = {0, 4, 7}. Both H and K have three elements, but they are not equal.
Therefore, none of the options (a, b, c) can be inferred solely based on the information provided about the group G, and its subgroups H and K.
