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I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ...
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:
View attachment 7993
View attachment 7994
In the above proof by Rotman we read the following:
" ... ... For the reverse inclusion let $$a \in \pi^{-1} \pi (S)$$, so that $$\pi (a) = \pi (s)$$ for some $$s \in S$$. It follows that $$as^{-1} \in \text{ ker } \pi = K$$ ... ... "Can someone please explain exactly how/why $$\pi (a) = \pi (s)$$ implies that $$as^{-1} \in \text{ ker } \pi = K$$ ... ... Peter
===========================================================================================***EDIT***
Just had some thoughts ... BUT ... unfortunately my logic does not seem to line up with Rotman's logic ...
My thoughts are as follows:
$$\pi (a) = \pi (s) $$
$$\Longrightarrow aK = sK$$
$$\Longrightarrow a = sk$$ for some $$k \in K$$ since $$a$$ must belong to $$sK$$ ... ... (is this a legitimate step ...)
$$\Longrightarrow s^{-1} a = k$$
$$\Longrightarrow as^{-1} = k$$
$$\Longrightarrow as^{-1} \in \text{ker } \pi = K$$ Is that correct?
BUT note ... logic is different from Rotman's set of steps ... ... what is Rotman's logic ...?Peter
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:
View attachment 7993
View attachment 7994
In the above proof by Rotman we read the following:
" ... ... For the reverse inclusion let $$a \in \pi^{-1} \pi (S)$$, so that $$\pi (a) = \pi (s)$$ for some $$s \in S$$. It follows that $$as^{-1} \in \text{ ker } \pi = K$$ ... ... "Can someone please explain exactly how/why $$\pi (a) = \pi (s)$$ implies that $$as^{-1} \in \text{ ker } \pi = K$$ ... ... Peter
===========================================================================================***EDIT***
Just had some thoughts ... BUT ... unfortunately my logic does not seem to line up with Rotman's logic ...
My thoughts are as follows:
$$\pi (a) = \pi (s) $$
$$\Longrightarrow aK = sK$$
$$\Longrightarrow a = sk$$ for some $$k \in K$$ since $$a$$ must belong to $$sK$$ ... ... (is this a legitimate step ...)
$$\Longrightarrow s^{-1} a = k$$
$$\Longrightarrow as^{-1} = k$$
$$\Longrightarrow as^{-1} \in \text{ker } \pi = K$$ Is that correct?
BUT note ... logic is different from Rotman's set of steps ... ... what is Rotman's logic ...?Peter
Last edited: