prove that the union of two subgroups is a subgroup iff one is contained in the other.

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Jeffrey Jordon · Accepted Answer

We will prove that(1)&amp;nbsp;K∪H&amp;nbsp;is a subgroup&amp;nbsp;&amp;nbsp;⟹&amp;nbsp;K⊂H∨H⊂KBy contradiction(H⊄K)∧(K⊄H)&amp;nbsp;⟺&amp;nbsp;{∃h∈H∖K∃k∈K∖H&amp;nbsp;Let&amp;nbsp;u=k∗h∈K∪Hnk∗h∈K∪H&amp;nbsp;⟹&amp;nbsp;{k∗h∈K&amp;nbsp;⟹&amp;nbsp;k−1∗k∗h∈K&amp;nbsp;⟹&amp;nbsp;h∈K&amp;nbsp;contradictionork∗h∈H&amp;nbsp;⟹&amp;nbsp;k∗h∗h−1∈H&amp;nbsp;⟹&amp;nbsp;k∈H&amp;nbsp;contradiction&amp;nbsp;So&amp;nbsp;K∪H&amp;nbsp;is a subgroup&amp;nbsp;&amp;nbsp;⟹&amp;nbsp;K⊂H∨H⊂K(2)&amp;nbsp;K⊂H∨H⊂K&amp;nbsp;⟹&amp;nbsp;&amp;nbsp;K∪H&amp;nbsp;is a subgroup is obvious

prove that the union of two subgroups is

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