Partition of a group
A partition of a group is an expression of the group as a set-theoretic union of subgroups, with pairwise trivial intersections.
The partition is said to be nontrivial if it uses more than one subgroup, or equivalently, if all the subgroups are proper.
Note that any nontrivial partition must involve at least three subgroups. Further information: Union of two subgroups is not a subgroup unless they are comparable
Not every group admits a nontrivial partition. Further information: group admitting a nontrivial partition