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I am currently focused on Chapter 1: Groups I ...

I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ...

Proposition 1.82 reads as follows:

https://www.physicsforums.com/attachments/7995

View attachment 7996

In the above proof by Rotman we read the following:

" ... ... To see that \(\displaystyle \Phi\) is surjective, let \(\displaystyle U\) be a subgroup of \(\displaystyle G/K\). Now \(\displaystyle \pi^{-1} (U)\) is a subgroup of \(\displaystyle G\) containing \(\displaystyle K = \pi^{-1} ( \{ 1 \} )\), and \(\displaystyle \pi ( \pi^{-1} (U) ) = U\) ... ... "My questions on the above are as follows:

**Question 1**How/why is \(\displaystyle \pi^{-1} (U)\) is a subgroup of \(\displaystyle G\) containing \(\displaystyle K\)? And further, how does \(\displaystyle \pi^{-1} (U) = \pi^{-1} ( \{ 1 \} )\) ... ... ?

**Question 2**

How/why exactly do we get \(\displaystyle \pi ( \pi^{-1} (U) ) = U\)? Further, how does this demonstrate that \(\displaystyle \Phi\) is surjective?

Help will be much appreciated ...

Peter