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.  ∀ ∈ H1∪H2

Let

H1 = {2n: n = Z} = {....-4, -2, 0, 2, 4, 6, ....}
H2 = {3n: n = Z} = {....-6, -3, 0, 3, 6, ....}
Since         0 =  2(0) ∈ H1, so H1 is a non-empty subset of G
Let a, b = H1. Then a = 2n1 and b = 2n2 for some n1, n2 ∈ Z. 
Now, a-b = 2n1 - 2n2 = 2(n1-n2) ∈ H1 
a-b ∈ H1 a, b ∈ H1 and so H1 is a subgroup of G. 
Similarly, H2 is a subgroup of G.
Also, H1∪H2 = {..... -6, -4, -3, -2, 0, 2, 3, 4, 6, ...} 
Now, 2, 3 ∈ H1∪H2
but 2-3=-1 ∉ H1∪H2 Thus, H1∪H2 is not a subgroup of G.

Youtube Video on this Theorem

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