Centrilizer of group
Answers
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.
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 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.
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.
hope this will help you.....
Please mark this answers as the brain list and follow me for more answers.......
The centralizer of an element z of a group G is the set of elements of G which commute with z,
C_G(z)={x in G,xz=zx}.
Likewise, the centralizer of a subgroup H of a group G is the set of elements of G which commute with every element of H,
C_G(H)={x in G, forall h in H,xh=hx}.
The centralizer always contains the group center of the group and is contained in the corresponding normalizer. In an Abelian group, the centralizer is the whole group.
example:
If G is a group, and H is a subgroup, then the normalizer of H in G is
NG(H)={g∈G∣g−1Hg=H},
and the centralizer is
CG(H)={g∈G∣gh=hg for all h∈H}.
It is easy to see that CG(H)⊆NG(H), but the converse need not hold.
For example, take G=S3, and let H={I,(1,2,3),(1,3,2)}.
What is CG(H)? It's the collection of all permutations that commute with I, with (1,2,3), and with (1,3,2). Since (1,2) does not commute with (1,2,3),
(1,2,3)(1,2)=(1,3)≠(2,3)=(1,2)(1,2,3),
then (1,2)∉CG(H). However, (1,2) does normalize H:
(1,2)−1I(1,2)(1,2)−1(1,2,3)(1,2)(1,2)−1(1,3,2)(1,2)=I∈H;=(1,3,2)∈H;=(1,2,3)∈H.
So (1,2)∈NG(H). Similarly, (1,3) and (2,3) are not in the centralizer, but are in the normalizer. H is contained in both.
For another example, take G=H=S3. Then the normalizer is all of G, because for every x,g∈G we have gxg−1∈G; but the centralizer is equal to the center (the set of things that commute with everything) and the center of G is just the identity.
MARK ME AS BRAINLIEST PLS