How does one find the center of the group G/Z(G)?
Center of G/Z(G)
1 Answers
The same way one finds the center in any group: you find all elements that commute with everything in $G/Z(G)$, that's $Z(G/Z(G))$.
The subgroup of $G$ that contains $Z(G)$ and corresponds to $Z(G/Z(G))$ is called the "second center of $G$", $Z_2(G)$.
In general, recall that the commutator of $x$ and $y$ is defined to be $[x,y] = x^{-1}y^{-1}xy$. The commutator is the unique element of $G$ such that $xy = yx[x,y]$ and $x$ and $y$ commute if and only if $[x,y]=1$.
It is not hard to see that if $N\triangleleft G$, then $gN\in Z(G/N)$ if and only if for every $x\in G$, $[g,x]\in N$ (this, because $gN$ commutes with $xN$ if and only if $1 = [gN,xN] = [g,x]N$, if and only if $[g,x]\in N$). So one can also describe the second center of $G$ as $Z_2(G) = \{ g\in G\mid [g,x]\in Z(G)\text{ for all }x\in G\}$ and then the center of $G/Z(G)$ equals $Z_2(G)/Z(G)$.
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0Thanks a lot for the answer. – 2011-03-18