Intereting Posts

An oddity in some linear equations
Intuition of Gronwall lemma
An overview of analysis
Associative ring with identity, inverses, divisors of zero and Artinianity
Unity in Partial Ring of Quotients $Q(R, T)$
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Alternating sum of multiple zetas equals always 1?
How does Mathematica solve $f(x)\equiv 0\pmod p$?
Difficulties in solving a PDE problem
Normal Intersection of Parabola
Number of ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$.
Sudoku puzzle with exactly 3 solutions
For $n \in \mathbb{N}$ $\lfloor{\sqrt{n} + \sqrt{n+1}\rfloor} = \lfloor{\sqrt{4n+2}\rfloor}$
An unusual symmetric inequality of trigonometric functions
This infinitely nested root gives me two answers $ \sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{…}}}}}} $

Possible Duplicate:

Prove that the center of a group is a normal subgroup

Suppose that $H$ is a normal subgroup of $G$. Prove that $C_{G}(H)$ is a normal subgroup of $G$, where $C_{G}(H)$ is the centralizer of $H$ in $G$.

I have proved that $C_{G}(H)$ is a subgroup but how do I prove that it is normal – is this not obvious by the definition of a centralizer?

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Let $x\in C_G(H)$, and $g\in G$ arbitrary. Then, as $xh=hx$ for all $h\in H$, and $H$ is normal, thus $g^{-1}hg\in H$, we have

$$gxg^{-1}\cdot h=gx(g^{-1}hg)g^{-1} = g(g^{-1}hg)xg^{-1} = h\cdot gxg^{-1} .$$

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