Math, asked by anisakabir56, 8 months ago

Theorem
Centre, Centeralizer and normalizer of an Abelian
group is group itself​

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Answered by 4442vickychoudhary
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Answer:

In mathematics, especially group theory, the centralizer (also called commutant[1][2]) of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S is the set of elements that satisfy a weaker condition. The centralizer and normalizer of S are subgroups of G, and can provide insight into the structure of G.

The definitions also apply to monoids and semigroups.

In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring R is a subring of R. This article also deals with centralizers and normalizers in Lie algebra.

The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.

Definitions Edit

Group and semigroup Edit

The centralizer of a subset S of group (or semigroup) G is defined as[3]

{\displaystyle \mathrm {C} _{G}(S)=\{g\in G\mid gs=sg{\text{ for all }}s\in S\}.}{\displaystyle \mathrm {C} _{G}(S)=\{g\in G\mid gs=sg{\text{ for all }}s\in S\}.}

If there is no ambiguity about the group in question, the G can be suppressed from the notation. When S = {a} is a singleton set, we write CG(a) instead of CG({a}). Another less common notation for the centralizer is Z(a), which parallels the notation for the center. With this latter notation, one must be careful to avoid confusion between the center of a group G, Z(G), and the centralizer of an element g in G, Z(g).

The normalizer of S in the group (or semigroup) G is defined as

{\displaystyle \mathrm {N} _{G}(S)=\{g\in G\mid gS=Sg\}.}{\displaystyle \mathrm {N} _{G}(S)=\{g\in G\mid gS=Sg\}.}

The definitions are similar but not identical. If g is in the centralizer of S and s is in S, then it must be that gs = sg, but if g is in the normalizer, then gs = tg for some t in S, with t possibly different from s. That is, elements of the centralizer of S must commute pointwise with S, but elements of the normalizer of S need only commute with S as a set. The same notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with the normal closure.

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