Normalizer of a subgroup

For the associated subgroup property, refer normalizer subgroup

You might be looking for the more general notion of: normalizer of a subset of a group

Symbol-free definition
The normalizer (normaliser in British English) of a subgroup in a group is any of the following equivalent things:


 * 1) The largest intermediate subgroup in which the given subgroup is normal.
 * 2) The set of all elements in the group for which the induced inner automorphism restricts to an automorphism of the subgroup.
 * 3) The set of all elements in the group that commute with the subgroup.

Definition with symbols
The normalizer of a subgroup $$H$$ in a group $$G$$, denoted as $$N_G(H)$$, is defined as any of the following equivalent things:


 * 1) The largest group $$K$$ for which $$H \le K \le G$$ and $$H$$ is normal in $$K$$.
 * 2) The set of all elements $$x$$ for which the map sending $$g$$ to $$xgx^{-1}$$ restricts to an automorphism of $$H$$.
 * 3) The set of all elements $$x$$ for which $$Hx = xH$$.

Inverse image of whole group
A subgroup is normal in the whole group if and only if its normalizer is the whole group. Thus the collection of normal subgroups can be thought of as the inverse image of the whole group under the normalizer map.

Iteration
The $$k$$-times iteration of normalizer is termed the $$k$$-hypernormalizer and a subgroup whose $$k$$-times hypernormalizer is the whole group is termed a $$k$$-hypernormalized subgroup. The condition of being $$k$$-hypernormalized is stronger than the condition of being $$k$$-subnormal.

Fixed-points
A subgroup of a group that is its own normalizer is termed a self-normalizing subgroup.

Textbook references

 * , Page 34 (definition in paragraph)
 * , Page 50 (formal definition; more generally defines normalizer of a subset), alternative definition given on Page 88, as a consequence of Exercise 31
 * , Page 47, Problem 16 (definition introduced in problem 13)
 * , Page 14, Remark 2
 * , Page 204, Point (3.7) (defines normalizer as stabilizer in terms of group action by conjugation on conjugacy class of subgroups)
 * , Page 38 (more generally defines normalizer of a subset)
 * , Page 82, Example 5.2.1(iv)
 * , Page 89 (definition in paragraph, under Examples)
 * , Page 219, Definition 4.2.5 (formal definition)