A subgroup is a subset 
of group elements of a group 
that satisfies the four group
requirements. 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 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 
.
