B. H. Neumann's lemma

Statement
If a group arises as a union of finitely many fact about::left cosets of (possibly same, possibly different) subgroups, then at least one of those is a coset of a fact about::subgroup of finite index.

Note that since a left coset of a subgroup is a fact about::right coset of one of its conjugate subgroups (specifically, $$gH = (gHg^{-1})g$$), and since conjugate subgroups have the same index in the whole group, we can replace left coset by right coset above.

Unions of subgroups

 * Union of two subgroups is not a subgroup unless they are comparable
 * Union of three subgroups is a subgroup implies they have index two and form a flower arrangement
 * Union of n subgroups is the whole group iff the group admits one of finitely many groups as quotient
 * There is no group that is a union of seven proper subgroups but not a union of fewer proper subgroups

Related facts in group theory

 * Directed union of subgroups is subgroup
 * Union of all conjugates is proper
 * Every group is a union of cyclic subgroups
 * Every group is a union of maximal among Abelian subgroups
 * Cyclic iff not a union of proper subgroups

The geometry of cosets

 * Coset containment implies subgroup containment
 * Nonempty intersection of cosets is coset of intersection