On group theory 3
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On group theory 3
Let \(\displaystyle{\left(G,\cdot\right)}\) be a group and \(\displaystyle{N}\) a normal subgroup of \(\displaystyle{\left(G,\cdot\right)}\), that is \(\displaystyle{N\trianglelefteq G}\) .
Prove that there exists a subgroup \(\displaystyle{K}\) of the group \(\displaystyle{Z(G/N)}\) such that:
\(\displaystyle{\left(Z(G)/Z(G)\cap N,\cdot\right)\simeq K\leq \left(Z(G/N),\cdot\right)}\).
Prove that there exists a subgroup \(\displaystyle{K}\) of the group \(\displaystyle{Z(G/N)}\) such that:
\(\displaystyle{\left(Z(G)/Z(G)\cap N,\cdot\right)\simeq K\leq \left(Z(G/N),\cdot\right)}\).
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- Community Team
- Posts: 426
- Joined: Mon Nov 09, 2015 1:52 pm
Re: On group theory 3
We give a solution :
Let \(\displaystyle{f:Z(G)\longrightarrow Z(G/N)\,,f(x)=x\,N}\) .
If \(\displaystyle{x\in Z(G)}\), then \(\displaystyle{x\,y=y\,x\,,\forall\,y\in G}\) .
We will prove that \(\displaystyle{x\,N\in Z(G/N)}\). Indedd,
\(\displaystyle{x\,N\cdot y\,N=(x\,y)\,N=(y\,x)\,N=y\,N\cdot x\,N\,,\forall\,y\,N\in G/N}\) .
So, the function \(\displaystyle{f}\) is well defined.
If \(\displaystyle{x\,,y\in Z(G)}\), then
\(\displaystyle{f(x\,y)=(x\,y)N=(x\,N)\,(y\,N)=f(x)\,f(y)}\), that is, \(\displaystyle{f}\) is homomorphism.
Finally,
\(\displaystyle{\begin{aligned} \rm{Ker}(f)&=\left\{x\in Z(G): f(x)=N\right\}\\&=\left\{x\in Z(G): x\,N=N\right\}\\&=\left\{x\in Z(G): x\in N\right\}\\&=Z(G)\cap N\end{aligned}}\)
According to the 1st Isomomorphism theorem, we get :
\(\displaystyle{\left(Z(G)/Z(G)\cap N,\cdot\right)\simeq K=Im(f)\leq \left(Z(G/N),\cdot\right)}\)
Let \(\displaystyle{f:Z(G)\longrightarrow Z(G/N)\,,f(x)=x\,N}\) .
If \(\displaystyle{x\in Z(G)}\), then \(\displaystyle{x\,y=y\,x\,,\forall\,y\in G}\) .
We will prove that \(\displaystyle{x\,N\in Z(G/N)}\). Indedd,
\(\displaystyle{x\,N\cdot y\,N=(x\,y)\,N=(y\,x)\,N=y\,N\cdot x\,N\,,\forall\,y\,N\in G/N}\) .
So, the function \(\displaystyle{f}\) is well defined.
If \(\displaystyle{x\,,y\in Z(G)}\), then
\(\displaystyle{f(x\,y)=(x\,y)N=(x\,N)\,(y\,N)=f(x)\,f(y)}\), that is, \(\displaystyle{f}\) is homomorphism.
Finally,
\(\displaystyle{\begin{aligned} \rm{Ker}(f)&=\left\{x\in Z(G): f(x)=N\right\}\\&=\left\{x\in Z(G): x\,N=N\right\}\\&=\left\{x\in Z(G): x\in N\right\}\\&=Z(G)\cap N\end{aligned}}\)
According to the 1st Isomomorphism theorem, we get :
\(\displaystyle{\left(Z(G)/Z(G)\cap N,\cdot\right)\simeq K=Im(f)\leq \left(Z(G/N),\cdot\right)}\)
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