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Let $G$ be a (not necessarily finite) group and $N\lhd G$. Show that if $N$ is finite then $C_G(N)$ has finite index in $G$. Further show that if $N$ is finite and $G/N$ is cyclic then the center $Z(G)$ has finite index.

These two are driving me crazy, and I simply cannot figure them out... Any help is appreciated

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2 Answers 2

up vote 6 down vote accepted

Since $N$ is normal, $G$ acts on $N$ by conjugation, giving a homomorphism from $G$ to $\rm{Aut}(N)$. The kernel of this map is exactly $C_G(N)$ so since $N$ only has a finite number of automorphisms, the index must be finite.

For the second one, we have $G = N\left< g \right>$ for some $g\in G$ (just take a generator of the quotient). Now, we have that $\left< g \right>$ acts on $N$ by conjugation, so the kernel of the corresponding map has finite index in $\left< g \right>$. But all the elements in this kernel commute with everything in $N$ and also with everything in $\left< g \right>$ and hence with everything in $G$, so the center of $G$ contains this kernel, which means that it must have finite index.

Here is a more general version: Let $G$ be a group with a finite normal subgroup $N$ and an abelian subgroup $H$ such that $G = NH$. Then $Z(G)$ has finite index in $G$. The proof is the same as above, since $H$ acts on $N$ by conjugation and the kernel is contained in $Z(G)$ and has finite index in $H$ which has finite index in $G$.

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I thought the OP's second claim might be wrong, but you made me light. +1 –  Babak S. Jan 3 '13 at 17:10
    
Another way to say this: By mapping $g$ to its conjugation action, there's a map from $C_G(N)$ to the group of automorphisms of $G$ fixing $N$. But any such automorphism descends to an automorphism of $G/N$. Any cyclic group has a finite automorphism group, so the image of the map $C_G(N) \to Aug(G/N)$ is finite, but its kernel is clearly the center of $G$. Since $C_G(N)$ had finite index, and $Z$ has finite index in $C_G(N)$, by the third isomorphism theorem, $Z$ has finite index in $G$. –  user54535 Jan 3 '13 at 18:32
    
Very clear. Thank you for your help –  Bey Jan 4 '13 at 21:40

For your first question, use $N/C$ theorem:

$(N/C)$ Theorem: If $G$ is a group and $N\subset G$, then $N_G(N)/C_G(N)$ is isomorphic to a subgroup of $\text{Aut}(N)$.

So since $N$ is normal in the group so the normalizer of it is $G$ and we have $G/C_G(N)\cong S\leq Aut(N)$ which is finite.

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Nice theorem to use here! +1 –  amWhy Feb 13 '13 at 0:04

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