Subgroup

No group is the union of two proper subgroups

For any group it is impossible to construct the entire group as the union two proper subgroups . group

Proof

Let be strict subgroups of such that . Clearly these subgroups cannot be identical, so without loss of generality assume that there exists some such that . For any it follows that . If then which is a contradiction. If then . Therefore for any also , hence and , contradicting the requirement that . Thus no group is the union of two proper subgroups.

For more than two proper subgroups it is possible. For example, consider the group , itself a subgroup of . In this case , which are all proper subgroups.

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