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.

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.