A subgroup is a subset of elements in the group that is itself a group, for example the set of all real numbers less zero is a subgroup of , and the set of all rational numbers (less zero) is similarly a subgroup of . The mathamatical notation for a subgroup is the same as that of a subset:
The trivial group is obviously a subgroup of all groups. Also a group is always its own subgroup. A simple group is one with only these two subgroups - one cannot find any set of elements smaller than the entire group except the trivial group that is a subset.