To determine whether H ∪ K is a normal subgroup of G for all normal subgroups H, K of every group G or not.
let's first define what a normal subgroup is.
What is a normal subgroup?
A subgroup N of a group G is a normal subgroup if and only if gNg^−1 = N for all g ∈ G. This implies that for any element g ∈ G and any element n ∈ N, the conjugate gng^-1 is still in N.Hence, if H and K are normal subgroups of a group G, then H ∪ K is not always a normal subgroup of G. For example, if we take the group G to be the Klein four-group V4, which has subgroups {e, a} and {e, b}, then H ∪ K = {e, a, b} is not a normal subgroup of G, since a^{-1}ba = b is not in H ∪ K. Therefore, H ∪ K is a normal subgroup of G for all normal subgroups H, K of every group G is a false statement, and we have just shown a counterexample.