MHB The Correspondence Theorem for Groups .... Rotman, Proposition 1.82 ....

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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:
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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
 
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Hi, Peter.

The reasoning in your EDIT section copied below is correct, including the step where you ask if $a=sk$ is legitimate (it is because $K$ is a (sub)group and so must contain the identity, which implies $a\in aK$).

Peter said:
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 ...?

Indeed, your reasoning is expressed a little different from Rotman's (with yours on the level of elements from $G$ throughout and with Rotman's given on the level of cosets and group homomorphisms), but it is equally valid and yields the same conclusion, well done!

Peter said:
Can someone please explain exactly how/why $$\pi (a) = \pi (s)$$ implies that $$as^{-1} \in \text{ ker } \pi = K$$ ... ...

For any group homomorphism, say $h:G_{1}\rightarrow G_{2},$ $h(g^{-1})=\left[h(g)\right]^{-1}$ (a short but good exercise to try for yourself if you're not already familiar with this fact). Hence, $\pi(a)=\pi(s)\Longrightarrow\left[\pi(a)\right]^{-1}=\left[\pi(s)\right]^{-1}.$

Now we compute: $\pi(as^{-1})=\pi(a)\pi(s^{-1})=\pi(a)\left[\pi(s)\right]^{-1}=\pi(a)\left[\pi(a)\right]^{-1}=\text{Identity in}~G/K=K.$
 
GJA said:
Hi, Peter.

The reasoning in your EDIT section copied below is correct, including the step where you ask if $a=sk$ is legitimate (it is because $K$ is a (sub)group and so must contain the identity, which implies $a\in aK$).
Indeed, your reasoning is expressed a little different from Rotman's (with yours on the level of elements from $G$ throughout and with Rotman's given on the level of cosets and group homomorphisms), but it is equally valid and yields the same conclusion, well done!
For any group homomorphism, say $h:G_{1}\rightarrow G_{2},$ $h(g^{-1})=\left[h(g)\right]^{-1}$ (a short but good exercise to try for yourself if you're not already familiar with this fact). Hence, $\pi(a)=\pi(s)\Longrightarrow\left[\pi(a)\right]^{-1}=\left[\pi(s)\right]^{-1}.$

Now we compute: $\pi(as^{-1})=\pi(a)\pi(s^{-1})=\pi(a)\left[\pi(s)\right]^{-1}=\pi(a)\left[\pi(a)\right]^{-1}=\text{Identity in}~G/K=K.$
Thanks GJA ...

... most helpful !

Peter
 
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