Suppose that $G$ is finite group with two normal subgroups $N$ and $K$ such that $K
Center of a finite group
-
0@Morad: This is what I essentially prove in my comment. I say essentially, because (assuming $L$ and $M$ are fixed transversals) $L$ might be $a, b, \ldots$ but then $M$ might have $an$ in place of $a$, where $n\in N$. However, $a$ and $an$ are equal mod $N$. – 2012-10-23
2 Answers
For showing that what @Nicky pointed, consider the Quaternion group $Q_8=\{+1,-1,+i,-i,+j,-j,+k,-k\}$ of order $8$. Let $K=\{+1,-1\}, N=\{+1,-1,+i,-1\}$ which are two normal proper subgroups of $Q_8$ such that $K
-
0"Groupie" Babak! +1 – 2013-02-12
My comments answer your question. I shall expand on them here.
If $K\leq N$ are both normal subgroups of a group $G$ with $Z(G/N)=M/N$ and $Z(G/K)=L/K$ then you are wanting to show that $L\subset M$. However, this doesn't really work. Suppose $G$ is the cyclic group of order $4$, with elements $\{0, 1, 2, 3\}$, and take $K=\{0, 2\}$ and $N=G$. Then $Z(G/K)=\{0K, 1K\}$ so we can take $M=\{0, 1\}$. On the other hand, $G/N$ is trivial, so we shall write $G/N=\{3N\}$, and so $L=\{3\}$. This contradicts your assertation.
This is, of course, silly. But entirely valid!
Basically, you need to take $L=\{h: hK\in Z(G/K)\}$ and $M=\{h: hN\in Z(G/N)\}$. Then your theorem works, as if $h\in L$ then
$\begin{align*} &hKgK=gKhK \:\forall\: g\in G\\ \Rightarrow &hgK=ghK\:\forall\: g\in G\\ \Rightarrow &hgh^{-1}g^{-1}\in K\:\forall\: g\in G\\ \Rightarrow &hgh^{-1}g^{-1}\in N\: \forall\: g\in G \: (as\: K\leq N)\\ \Rightarrow &h\in M\\ \Rightarrow &L\subset M \end{align*}$
as required.
On the other hand, I am not sure if you were but you could have been using the correspondence theorem to take $L$ and $M$ as subgroups of $G$ which contain $K$ and $N$ respectively. The above proof still works in this case, by the uniqueness of $L$ and $M$.
-
1@Morad: Can you explain what you think happens when $Z(N)=1$? My point is, if $N$ and $K$ are both proper subgroups of $G$ then we can (often? always?) take transversals for $G/K$ and $G/N$ which intersect trivially. $Z(G/N)$ is just a subset of the transversal of $G/N$, and so intersects the transversal for $G/K$ trivially. – 2012-10-29