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Take a group $G$ and a subgroup $H$. Let's define the set of left cosets of H as: $$S=\{aH | a\in G\}$$ It's pretty clear to me we can define a left action from G to S like this:$$g(aH)=gaH$$ This basically permutes the cosets while G is acting on them. However, I need a confirmation with a simpler question (probably trivial for this space). The way in which we defined the left cosets set, actually suggests we are letting elements of $G$ running on the subgroup $H$, obtaining all cosets.

I mean, it sounds a bit if we could identify the cosets as orbits of this "action". The wrong part affecting this "feeling" seems to me this cannot be called an action at all, technically.

An action is an homomorphism from G to Perm(S), where S is the "target" set the group acts onto.

So here we are somewhat letting G acting on H producing new cosets, but the result is not any permutation of H. So this is not a homomorphism.

I've googled the net but no results for "group left action on a subgroup".

So the only left action I can see related to this is the one already considering the whole set of left cosets as target.

Am I correct in solving my doubts?

thanks for your support

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    $\begingroup$ I don't really understand what you're looking for, but if $H$ acts on $G$ on the right by $G\times H\to G, (g,h)\mapsto gh$, then the orbits of this right action are precisely the cosets $gH$. Now if you don't like right actions, you can make this a left action by $H\times G\to G, (h,g)\mapsto gh^{-1}$, it doesn't change much. So $G/H$ is the set of orbits of $G$ under the action of $H$ $\endgroup$ – Max Jan 29 at 11:18
  • $\begingroup$ What you wrote is known to me. But , no matter left or right action, you are suggesting an action of H on G. I'm trying to see if we could get the cosets with the opposite action, i.e. from the G to the subgroup H. But your suggestion makes so this is probably what I'm looking for. I was looking for G acting on H generating the cosets. But that "acting" is not an "action" indeed. It's a sort of "generating f()" creating cosets, but it cannot be turned into any homomoprhism. I hope makes my idea more clear. $\endgroup$ – riccardoventrella Jan 29 at 11:24
  • $\begingroup$ @Max, I really think your answer showed to me what I really had in mind. Initially it was a bit unnatural using right action to get left cosets, but the more I think of the more it's really powerful. So, why not ranking higher your comment to an answer, in order to let you benefit from my selection? Thanks $\endgroup$ – riccardoventrella Jan 29 at 12:25
  • $\begingroup$ I don't understand, but if you want I can copy my comment into an answer ? $\endgroup$ – Max Jan 29 at 13:01
  • $\begingroup$ Yes, I'm not an expert of this space but you comment answered to my question, but I cannot assign to you any points, or select it as official answer. I don't know if it's the right procedure, but I think it could worth letting you benefit from that. Do as you wish, I don't want to push you to do actions outside this space policy. Thanks $\endgroup$ – riccardoventrella Jan 29 at 13:04
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I don't really understand what you're looking for, but if $H$ acts on $G$ on the right by $G\times H\to G,(g,h)\mapsto gh$, then the orbits of this right action are precisely the cosets $gH$. Now if you don't like right actions, you can make this a left action by $H\times G\to G,(h,g)\mapsto gh^{−1}$, it doesn't change much. So $G/H$ is the set of orbits of $G$ under the action of $H$

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  • $\begingroup$ To strengthen this: groupprops.subwiki.org/wiki/… "The left cosets are the orbits of G under the action of H on the right by multiplication. G has a left action on the left coset space by left multiplication." $\endgroup$ – riccardoventrella Jan 29 at 15:06

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