Suppose that $G$ is finite group with two normal subgroups $N$ and $K$ such that $K
Center of a finite group
1
$\begingroup$
abstract-algebra
group-theory
-
3$G/K$ is not necessarily a *subset* of $G/N$, so what do you mean precisely? If $Z(G/K)=L/K$ and $Z(G/N)=M/N$, do you want to know if $L \subset M$? – 2012-10-23
-
0If $hK\in Z(G/K)$ then are you wanting to prove that $hK\in Z(G/K)$? Well, if $hK\in Z(G/K)$ then $[g, h]\in K\leq N$ for all $g\in G$. So, we're done. – 2012-10-23
-
0@user1729 I think you mean "wanting to prove that $hN \in Z(G/N)$" right? – 2012-10-23
-
0@NickyHekster Yup, thanks. – 2012-10-23
-
0Thanks for all comments. Yes Nicky I want to know if $L$ is subset of $M$? – 2012-10-23
-
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