A *subgroup* is a sub*set* 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:

or

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.

Robert G. Brown 2017-07-11