I Differences between left vs right actions in some group theory questions

elias001
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The following is taken from A first Course in Abstract Algebra Rings, Groups, and Fields Third Edition by Anderson and Feil.##\\\\##



(Assumed exercise and example) ##\\\\##

22.15 In this problem we consider a particular important example of a group endomorphism. Suppose ##G## is a group and ##g\in G##. Define the function ##\varphi:G\to G## by setting ##\varphi(h)=ghg^{-1}\\\\##

(a) Prove that ##\varphi## is a group isomorphism##\\\\##

(b) What can you say about ##\varphi## if the group is abelian? What if ##g\in Z(g),## the center of ##G.\\\\##

Example 28.4##\\\\##

Let ##G## act on ##G## by conjugation given by ##g(x)=gxg^{-1};## see Exercise 22.15. In that exercise you show that the function ##g(x)## is a bijection (that is, a permutation of ##G##). We claim also that this action is homomorphic: That is, we claim that ##g(h(x))=(gh)(x).\\\\##

Exercise Question:##\\\\##

6. Let ##S## be a subgroup of the group ##G.## We wish to consider ##G## acting on the right cosets of ##S## by right multiplication. That is, ##g(Sh)=Shg.## Show that this does not work in general. That is, ##G## is not a group of permutations of the right cosets of ##S.## What goes wrong here? ##\\\\##

7. We wish to let ##G## act on ##G## by conjugation ##g(x)=g^{-1}xg.## Note this is different from conjugation we used in Example 28.4. Show that this does not work in general. What goes wrong here?##\\\\##

8. We ask the same question as the previous exercise, where we try to have ##G## act on its set of subgroups by the conjugation, ##g(S)=g^{-1}Sg.## What goes wrong here?##\\\\##

(c) Show that the order of ##h## is equal to the order of ##\varphi(h)=ghg^{-1}.\\\\##

Questions: I have some quick questions. For the Exercises Questions 6, 7,8; why can't right action not work in those three exercises? Is it the way $g$ is defined in all three questions or does it have to do with how or what $G$ acts on? I thought there is symmetry in the definition of group actions. Meaning if we know how to define it for right action, it should be the same for left action? Am I missing something?
 
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elias001 said:
Questions: I have some quick questions. For the Exercises Questions 6, 7,8; why can't right action not work in those three exercises? Is it the way $g$ is defined in all three questions or does it have to do with how or what $G$ acts on? I thought there is symmetry in the definition of group actions. Meaning if we know how to define it for right action, it should be the same for left action? Am I missing something?
This is the point of these exercises. There is a difference between left and right action. I don't have the book, but I am sure that the definitions are given. Try to do the exercises, if you just see the answer you are not going to grok it as well as when you do the work.
 

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