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Home » Math Homework Help » Algebra Homework Help » Normalizer or Centalizer
Normalizer or Centalizer
The set of all those elements of G, which commute with a given element a is called the normalizer of a and will be denoted by N (a).

Thus, N (a) = {x G : ax = xa}.

We have the following results.

(i) Since, ex = xe ∀ x G ; so N (e) = G.

i.e. the normalizer of the identity element is the whole group G.

(ii) If a G and G is abelian, then ax = xa, ∀ x G.

Consequently, N (a) = G.

i.e. the normalizer of every element of an abelian group G is always a sub-group of G.

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