Union of three proper subgroups is the whole group implies they have index two and form a flower arrangement

Name
This result is often termed Scorza's theorem since it was first proved in a paper by Scorza.

Statement
Suppose $$G$$ is a group and $$H_1, H_2, H_3$$ are proper subgroups of $$G$$ such that the union is the whole group $$G$$:

$$G = H_1 \cup H_2 \cup H_3$$

Then each $$H_i$$ has index two in $$G$$, and they form a flower arrangement of subgroups:

$$H_1 \cap H_2 = H_1 \cap H_3 = H_2 \cap H_3$$.

Further, this intersection is a normal subgroup of $$G$$ and the quotient is isomorphic to the Klein four-group.

Related facts

 * Union of two subgroups is not a subgroup unless they are comparable
 * B.H.Neumann's lemma
 * There is no group that is a union of seven proper subgroups but not a union of fewer proper subgroups
 * Union of n proper subgroups is the whole group iff the group admits one of finitely many groups as quotient