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

From Groupprops

## Name

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

## Statement

Suppose is a group and are proper subgroups of such that the union is the whole group :

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

.

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