Normalizer

The normalizer is a term from the mathematical branch of group theory .

definition

Let it be a group and a non-empty subset of . The normalizer of in is defined as ${\displaystyle G}$ ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle U}$ ${\displaystyle G}$

${\displaystyle N_{G}(U):=\left\{g\in G\mid gUg^{-1}=U\right\}}$.

Here is , according to the definition of the complex product . ${\displaystyle gUg^{-1}=\left\{gug^{-1}\mid u\in U\right\}}$

In other words: the normalizer consists of those for which it holds that under conjugation with is invariant. (These elements are said to normalize. ) ${\displaystyle N_{G}(U)}$ ${\displaystyle g\in G}$ ${\displaystyle U}$ ${\displaystyle g}$ ${\displaystyle U}$

Note that it is only required that the whole remains fixed, so in general this applies to individual elements and absolutely ; but it always applies . ${\displaystyle U}$ ${\displaystyle u\in U}$ ${\displaystyle g\in N_{G}(U)}$ ${\displaystyle gug^{-1}\neq u}$ ${\displaystyle gug^{-1}\in U}$

properties

• The normalizer is a subset of . ${\displaystyle G}$
• The index of the normalizer gives the number of different conjugates of the set , i.e. H. . ${\displaystyle N_{G}(U)}$ ${\displaystyle gUg^{-1}}$ ${\displaystyle U}$ ${\displaystyle |\{gUg^{-1}\mid g\in G\}|=[G:N_{G}(U)]}$
• A subgroup is always a normal subgroup in its normalizer . More precisely: is the largest subgroup of , in terms of inclusion , in which is normal. ${\displaystyle U}$ ${\displaystyle N_{G}(U)}$ ${\displaystyle N_{G}(U)}$ ${\displaystyle G}$ ${\displaystyle U}$
• A subgroup is a normal subgroup in if and only if its normalizer is whole . ${\displaystyle G}$ ${\displaystyle G}$
• One can also introduce the normalizer like this:
  Be a group. You leave on the power set of by conjugation operate . Then the stabilizer of this operation for a given subset of is just the normalizer of that subset. ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$

example

Let it be the group of invertible matrices (with real entries) for a natural number . Next is the subgroup of diagonal matrices . Then the normalizer of is in the group of matrices in which exactly one entry is not equal to zero in each row and in each column. The quotient is isomorphic to the symmetric group . ${\displaystyle G}$ ${\displaystyle n\times n}$ ${\displaystyle n}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle N_{G}(U)/U}$ ${\displaystyle S_{n}}$

Related terms

If one demands that element-wise is invariant under the conjugation with group elements, one gets the stronger concept of the centralizer . The centralizer is a normal divider in the respective normalizer. ${\displaystyle U}$ ${\displaystyle Z_{G}(U)}$