Editing Centralizer and normalizer (section)

==Example==
Consider the group 
:<math>G = S_3 = \{[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\}</math> (the symmetric group of permutations of 3 elements).

Take a subset <math>H</math> of the group <math>G</math>: 

:<math>H = \{[1, 2, 3], [1, 3, 2]\}. </math>

Note that <math>[1, 2, 3]</math> is the identity permutation in <math>G</math> and retains the order of each element and <math>[1, 3, 2]</math> is the permutation that fixes the first element and swaps the second and third element.

The normalizer of <math>H</math> with respect to the group <math>G</math> are all elements of <math>G</math> that yield the set <math>H</math> (potentially permuted) when the element conjugates <math>H</math>.
Working out the example for each element of <math>G</math>:

:<math>[1, 2, 3]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [1, 3, 2]\} = H</math>; therefore <math>[1, 2, 3]</math> is in the normalizer <math>N_G(H)</math>.
:<math>[1, 3, 2]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [1, 3, 2]\} = H</math>; therefore <math>[1, 3, 2]</math> is in the normalizer <math>N_G(H)</math>.
:<math>[2, 1, 3]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [3, 2, 1]\} \neq H</math>; therefore <math>[2, 1, 3]</math> is not in the normalizer <math>N_G(H)</math>.
:<math>[2, 3, 1]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [2, 1, 3]\} \neq H</math>; therefore <math>[2, 3, 1]</math> is not in the normalizer <math>N_G(H)</math>.
:<math>[3, 1, 2]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [3, 2, 1]\} \neq H</math>; therefore <math>[3, 1, 2]</math> is not in the normalizer <math>N_G(H)</math>.
:<math>[3, 2, 1]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [2, 1, 3]\} \neq H</math>; therefore <math>[3, 2, 1]</math> is not in the normalizer <math>N_G(H)</math>.

Therefore, the normalizer <math>N_G(H)</math> of <math>H</math> in <math>G</math> is <math>\{[1, 2, 3], [1, 3, 2]\}</math> since both these group elements preserve the set <math>H</math> under conjugation.

The centralizer of the group <math>G</math> is the set of elements that leave each element of <math>H</math> unchanged by conjugation; that is, the set of elements that commutes with every element in <math>H</math>. 
It's clear in this example that the only such element in S<sub>3</sub> is <math>H</math> itself ([1, 2, 3], [1, 3, 2]).