B.H.Neumann's lemma

Statement
If a group $$G$$ can be written as a union of left cosets:

$$G = g_1H_1 \cup g_2H_2 \cup \dots \cup g_nH_n$$

then $$G$$ is the union of those $$g_iH_i$$ for which $$H_i$$ is a subgroup of finite index in $$G$$. In other words, the cosets of subgroups of infinite index are redundant.

Facts about unions of subgroups

 * 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