Normalizer of a subset of a group - Revision history

Vipul: New page: ==Definition== Let $G$ be a group and $S$ be a subset of $G$ . The '''normalizer''' ('''normaliser''') of $S$ in $G$
2008-11-02T13:13:11Z

New page: ==Definition== Let <math>G</math> be a group and <math>S</math> be a subset of <math>G</math>. The '''normalizer''' ('''normaliser''') of <math>S</math> in <math>G</...

New page
==Definition==

Let <math>G</math> be a group and <math>S</math> be a [[subset of a group|subset]] of <math>G</math>. The '''normalizer''' ('''normaliser''') of <math>S</math> in <math>G</math>, denoted <math>N_G(S)</math> is defined as:

<math>N_G(S) := \{ g \in G \mid gSg^{-1} = S \}</math>.

Equivalently, it is the isotropy of <math>S</math> under the action of <math>G</math> on the set of subsets of <math>G</math> by conjugation.

We typically use the term '''normalizer''' for [[normalizer of a subgroup]], i.e., where the subset we start with is a [[subgroup]] of <math>G</math>.

==Facts==

* [[Group acts on set of subsets by conjugation]]