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

From Groupprops
Jump to: navigation, search

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