A subgroup is a subset 
of group elements of a group 
that satisfies the four group requirements under the inherited
group operation. It must therefore contain the
identity element. "
is a subgroup of 
" is written 
, or sometimes 
(e.g., Scott 1987, p. 16).
The order of any subgroup of a finite group of order 
must be a divisor of 
.
A subgroup 
of a group 
that does not include the entire group 
itself is known as a proper
subgroup, denoted 
or 
.
