Testwiki:Reference desk/Archives/Mathematics/2012 November 19

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November 19

Normal subgroups and bijection

Let $\varphi :G\to G^{\prime }$ be a surjective homomorphism. Suppose you have some subgroup $H^{\prime }$ of $G^{\prime }$ and define $H:={\varphi }^{-1}(H^{\prime })$. Let $N$ be a subgroup of $G$ containing $H$ and $N^{\prime }$ be a normal subgroup of $G^{\prime }$ containing $H^{\prime }$. Show that there is a bijection between $N$ and $N^{\prime }$.--AnalysisAlgebra (talk) 22:27, 19 November 2012 (UTC)

But what you've stated is clearly false. Sławomir Biały (talk) 00:06, 20 November 2012 (UTC)

I TOTALLY misunderstood the question. I need to show that there is a bijection between the set of subgroups of G and the set of subgroups of G' . I'm not sure if they need to be normal or not.--AnalysisAlgebra (talk) 08:46, 20 November 2012 (UTC)

Subgroups containing H and H' respectively, that is.--AnalysisAlgebra (talk) 08:48, 20 November 2012 (UTC)

They don't need to be normal. Start by showing that if $\varphi (x)=\varphi (y)$, then $x\in N\leftrightarrow y\in N$. This depends on $H\subseteq N$. Then use that to show that ${\varphi }^{-1}$ is the desired bijection.--149.148.254.207 (talk) 09:53, 20 November 2012 (UTC)

Hem. That's harder than it looks. You can get $\varphi (x)=\varphi (y)⟹(x\in H⟺y\in H)$. How does the result follow? How do you use that $\varphi$ is surjective?--AnalysisAlgebra (talk) 17:31, 20 November 2012 (UTC)

Hint:consider $\varphi (x^{-1}y)$.

Surjectivity isn't important; since the image of $\varphi$ is a subgroup, you could just replace $G^{\prime }$ with image$\varphi$.--80.109.106.49 (talk) 18:17, 20 November 2012 (UTC)

Yes, $x^{-1}y$ is in the kernel of $\varphi$. So what?--AnalysisAlgebra (talk) 18:46, 20 November 2012 (UTC)

What's the relationship between the kernel of $\varphi$ and H?--80.109.106.49 (talk) 19:02, 20 November 2012 (UTC)

All I can think of is $\ker \varphi$ is a suubgroup of $H$; did you have something else in mind? How does it relate to $N$?--AnalysisAlgebra (talk) 20:43, 20 November 2012 (UTC)

Since N contains H, this tells you that $x^{-1}y\in N$.--80.109.106.49 (talk) 20:59, 20 November 2012 (UTC)