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Let  be a group, and let
 . The normalizer of  in  , written  , is the set
A subgroup  of  is said to be self-normalizing if
 .
 is always a subgroup of  , as it is the stabilizer of  under the action
 of  on the set of all subsets of  (or on the set of all subgroups of  , if  is a subgroup).
If  is a subgroup of  , then
 .
If  is a subgroup of  , then  is a normal subgroup of  ; in fact,  is the largest subgroup of  of which  is a normal subgroup. In particular, if
 is a subgroup of  , then  is normal in  if and only if  .
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