Let $N,M$ be normal subgroups of $G$ with $N\cap M=\{e\}$. Prove that $M\subset C_{G}(N)$ and $N\subset C_{G}(M)$.

Question

First consider the following definition:

Let $G$ be a group and $H$ a subgroup of $G$. The center: $$ C_{G}(H)=\{g\in G\,:\,gh=hg,\,\forall h\in H\}$$

Now I'm trying to prove the following theorem:

Let $N,M$ be normal subgroups of $G$ so $N\cap M=\{e\}$.

Prove that $M\subset C_{G}(N)$ and $N\subset C_{G}(M)$.

What I tried:

We know that $N\triangleleft G$ so there is $n_{0}\in N$ so for every $n\in N$ and every $g\in G$, $n_{0}=g^{-1}ng$. We can convert it to be: $n=gn_0g^{-1}$. We also know that $M\triangleleft G$ so there is $m_{0}\in M$ so $m=gm_0g^{-1}$.

We can see that:

$$ nm=(gn_0g^{-1})(gm_0g^{-1})=g(n_0m_0)g^{-1}\\ mn=(gm_0g^{-1})(gn_0g^{-1})=g(m_0n_0)g^{-1}$$

But how I continue from here? As I understand I need to show that $nm=mn$.

Answer 1

First of all I'm not so sure about your definition of a normal subgroup. If $N\triangleleft G$ then what you know is that for any $g\in G,n\in N$ we have $gng^{-1}\in N$. It doesn't imply that $gng^{-1}$ is the same element for any $g$ and $n$. If anything, $geg^{-1}=e$ for any $g\in G$.

Now about the problem itself. Let $m\in M, n\in N$. We want to show that $mn=nm$. Now, because $N\triangleleft G$ we know that $mnm^{-1}\in N$. Since $n^{-1}\in N$ we conclude that $mnm^{-1}n^{-1}\in N$. On the other hand we also know that $M\triangleleft G$, so $nm^{-1}n^{-1}\in M$. Also $m\in M$ and hence $mnm^{-1}n^{-1}\in M$. So we got that $mnm^{-1}n^{-1}\in M\cap N=\{e\}$. Hence $mnm^{-1}n^{-1}=e$ and that implies $mn=nm$.

Answer 2

Observe that

$$K\lhd G\iff [K,G]=1$$

So apply this:

$$M\lhd G\iff [M,G]=1\;,\;\;\text{and in particular}\;\;\forall\,m\in M\,,\,n\in N: [m,n]=1\implies$$

$$m^{-1}n^{-1}mn=1\implies mn=nm\implies M\subset C_G(N)$$

Answer 3

It is useful to see $[M,N] := \langle mnm^{-1}n^{-1} |m\in M$ $ n\in N\rangle $. Shall show this is the trivial subgroup. $mnm^{-1} \in N$ and hence $mnm^{-1}n^{-1}\in N$ but then again $nm^{-1}n^{-1}\in M$ and hence $mnm^{-1} \in M$. So each generator is in $M\cap N=e$. Thus every element of $M$ commutes with every element if $N$. This solves your problem.