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We discussed Group Actions in my undergraduate Modern Algebra class today. I understand the definition and example we went over in lecture, but the problem set is proving difficult.

If I want to prove that a right group action $X \times G\to$ is equivalent data as a homomorphism $G$ op $\to \text{Aut}(X)$, how do I go about doing that? I know that a left group action on $X$ is same as ("induces") a right group action of $G$ op, but how do I link that back to automorphism?

My second question is about dihedral groups. If $G = D_6$, can any right group action on $X$ be faithful or transitive if $|X|=7$? I think $|X|=6$ must hold, but I'm not sure why. It goes on to ask about the possible values of $|X/G|$ for any right group action on $X$. What would that relationship imply?

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Well, question 1 is a little funny. There are two, in my mind, completely different things going on. The first is the difference between a left action and a right action, and how to convert between a left action of $G$ and a right action of $G^{op}$. You understand that perfectly well already.

The second is that a left action of $G$ on a set $X$ is the same as a homomorphism $G \to \text{Aut}(X)$. Do you know how to show this? (Hint: given an element $g$ and an action, you want a map of sets from $X$ to itself, so...)

Then you can combine these two to prove the exercise the way it is stated.

As for question 2, have you learned the orbit-stabilizer theorem?

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  • $\begingroup$ 2.) I have not learned the orbit-stabilizer theorem. I just looked it up, but I'm not quite sure what it means. Or, at least, how to use it. Something called Burnside's Lemma is discussed in conjunction to the Theorem-- is that what I'm supposed to use? $\endgroup$ – user113336 Dec 5 '13 at 1:57
  • $\begingroup$ And as for 1.), I'm sorry, but I don't understand your hint very well. If g is an element in G, how can it be part of an automorphism of X? $\endgroup$ – user113336 Dec 5 '13 at 2:04
  • $\begingroup$ Given $g \in G$ consider the map $X \to X$ given by $x \mapsto gx$. $\endgroup$ – hunter Dec 5 '13 at 9:34
  • $\begingroup$ without orbit-stabilizer (which is much easier than Burnside's Lemma) I don't know how you're supposed to do 2. $\endgroup$ – hunter Dec 5 '13 at 9:34
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Try "An introduction to the theory of groups/Joseph Rotman"- Theorem 3.18 - page_55
Or Rorman's "Advanced Modern Algebra" book page_99

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