The normal closure of a subset 
of a group 
is the smallest normal subgroup
of 
containing 
. It is generated by all conjugates 
with 
and 
, and is denoted 
(Brown 1982). The quotient 
imposes the relations 
for all 
.
For a singleton containing the identity element, the normal closure is trivial. For a group presentation, the normal closure of its relators in the free group gives the kernel of the map onto the presented group.
