Problem regarding a homomorphic mapping

Question

I can't understand this:

suppose we have a homomorphism $\phi :G \to G'$ such that it induces these two maps :

$$\phi_*:\tau\to \tau'$$ ,a map from subgroups of G to subgroups of G'

s.t $\phi_*(H)=\phi(H)$ and similarly,$$\phi^*:\tau'\to \tau$$ ,a map from subgroups of G' to subgroups of G. s.t $\phi^*(H)=\phi^{-1}(H)$

If these two maps aren't inverses of each other:
What I can't understand is this. we've
$$\phi_*\phi^*(H')=H'\cap im\phi$$
$$\phi^*\phi_*(H)=\langle H,Ker \phi \rangle$$ i.e smallest subgroup containing $H$ and $Ker \phi$

kindly help me with this.

It can only do good to your question (and eagerness to answer) to mention explicitly what is meant by $\tau$ and $\tau '$ — drhab, Oct 18 '14 at 08:37

drhab · Accepted Answer · 2014-10-20T16:03:32.217

2

$$y\in\phi_{*}\left(\phi^{*}\left(H\right)\right)\iff y=\phi\left(x\right)\text{ for some }x\in\phi^{*}\left(H\right)\iff $$$$y=\phi\left(x\right)\text{ for some }x\in G\text{ with }\phi\left(x\right)\in H\iff y\in H\wedge y\in\operatorname{im}\phi$$

and:

$$x\in\phi^{*}\left(\phi*\left(H'\right)\right)\iff\phi\left(x\right)\in\phi_{*}\left(H'\right)\iff$$$$\phi\left(x\right)=\phi\left(a\right)\text{ for some }a\in H'\iff x-a\in\text{ker}\phi\text{ for some }a\in H'\iff$$$$ x=a+b\text{ for some }a\in H'\text{ and some }b\in\operatorname{ker}\phi$$

edited Oct 20 '14 at 16:03

answered Oct 20 '14 at 14:50

drhab

151,093

thanks a lot for explanation.... – kittuu Oct 20 '14 at 17:59
you are very welcome. – drhab Oct 20 '14 at 18:19

Problem regarding a homomorphic mapping

1 Answers1