Normalizer of a subgroup

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This article defines a subgroup operator related to the subgroup property normal subgroup. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

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

Definition

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.

Related subgroup properties

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.

References

Textbook references

  • Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, More info, Page 34 (definition in paragraph)
  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 50 (formal definition; more generally defines normalizer of a subset), alternative definition given on Page 88, as a consequence of Exercise 31
  • Topics in Algebra by I. N. Herstein, More info, Page 47, Problem 16 (definition introduced in problem 13)
  • Algebra by Serge Lang, ISBN 038795385X, More info, Page 14, Remark 2
  • Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, More info, Page 204, Point (3.7) (defines normalizer as stabilizer in terms of group action by conjugation on conjugacy class of subgroups)
  • A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info, Page 38 (more generally defines normalizer of a subset)
  • An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, More info, Page 82, Example 5.2.1(iv)
  • Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, More info, Page 89 (definition in paragraph, under Examples)
  • A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, More info, Page 219, Definition 4.2.5 (formal definition)