Publish On: 2019-05-10

# Ben joy

Total Post: 542

## Question: What is a normalizer or centralizer

Reply On: 2013-10-07

# Mjay Jollay

Total Post: 0

## ANS: What is a normalizer or centralizer

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.

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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