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"the union of two sub-groups neednot be a sub-group".can some-body prove without using counter example ...

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3Z and 2Z are subgroups of (Z,+) but the union has elements 2 and 3. But 2+3=5∉3Z∪2Z. (Z,+) is cyclic, and hence abelian, and hence all subgroups of (Z,+)are normal. But the union of these 2 subgroups is not even a subgroup.

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