The Union of two subgroups of a group G may not be a subgroup of G
Theorem: The Union of two subgroups of a group G may not be a subgroup of G Proof: Let G = Z be the additive group of integers. ∀ ∈ H 1 ∪H 2 Let H1 = {2n: n = Z} = {....-4, -2, 0, 2, 4, 6, ....} H2 = {3n: n = Z} = {....-6, -3, 0, 3, 6, ....} Since 0 = 2(0) ∈ H 1 , so H 1 is a non-empty subset of G Let a, b = H 1 . Then a = 2n 1 and b = 2n 2 for some n 1 , n 2 ∈ Z. Now, a-b = 2n 1 - 2n 2 = 2(n 1 -n 2 ) ∈ H 1 a-b ∈ H 1 a, b ∈ H1 and so H1 is a subgroup of G. Similarly, H2 is a subgroup of G. Also, H 1 ∪H 2 = {..... -6, -4, -3, -2, 0, 2, 3, 4, 6, ...} Now, 2, 3 ∈ H 1 ∪H 2 but 2-3=-1 ∉ H 1 ∪H 2 Thus, H 1 ∪H 2 is not a subgroup of G. Youtube Video on this Theorem