There is no group that is a union of seven proper subgroups but not a union of fewer proper subgroups

History
This result was conjectured by Cohn and proved by Tomkinson, and is sometimes termed Tomkinson's theorem.

Statement
There is no group $$G$$ that can be written as the union of seven proper subgroups but cannot be written as a union of fewer proper subgroups.

Related facts

 * Union of two subgroups is not a subgroup unless they are comparable
 * Union of three proper subgroups is the whole group implies they have index two and form a flower arrangement
 * B.H.Neumann's lemma