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Given: $|G|=n$, $H$ is a subgroup of $G$ and $|G/H|=k$, where $n$ does not divide k!.

WTP: The left action map on $G/H$ has a nontrivial kernel.

I have not put the entire problem I am trying to work on here, just the part I am stuck on. I think this should be true, but I am having trouble showing nontriviality. I know if we take an element of G/H then the kernel of the map are all elements a such that $agH=gH$, but I cannot prove that there are any $a$'s that will do this other than the identity. Can anyone help, or tell me if this is not true?

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    $\begingroup$ Work this out with $S_3$ and a subgroup of order $2$. $\endgroup$
    – John Brevik
    Apr 20, 2015 at 3:17
  • $\begingroup$ The kernel are all elements $a$ such that $agH=gH$ for every $g$, i.e., the intersection of the stabilizers. $\endgroup$
    – Dustan Levenstein
    Apr 20, 2015 at 3:18
  • $\begingroup$ I'm sorry I mistyped, n should not divide k!, obviously it does not divide k,k-1,k-2,...,1. I have modified the question. $\endgroup$
    – Math Student
    Apr 20, 2015 at 3:22

1 Answer 1

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Hint: The left action map of $G$ on $G/H$ induces a homomorphism $\phi: G \to S(G/H) \simeq S_k$, given by $a \mapsto \{gH \mapsto agH\}$.

The kernel of the action is the kernel of this homomorphism. If the kernel were trivial, then $\phi$ would be injective, giving an embedding of $G$ as a subgroup of $S_k$. Why can't this happen?

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  • $\begingroup$ I think the homomorphism I am looking at is G/H on phi(G/H), would that be any different? Also, I have figured out what the kernel would be, but I am having difficulty proving the kernel is necessarily nontrivial, which I would assume follows from the second given, that n does not divide k!. Do you know how to prove nontriviality? Or maybe this is an obvious fact? As stated above, the overall question is actually a lot more than what I wrote, this is just one step of it that I am having trouble with. $\endgroup$
    – Math Student
    Apr 20, 2015 at 3:38
  • $\begingroup$ No idea what you mean by "I think the homomorphism I am looking at is..."; remember that $G/H$ is not necessarily a group. The induced homomorphism I speak of is given by $a \mapsto \{gH \mapsto agH\}$. What can be said about this homomorphism if the kernel is trivial? $\endgroup$
    – Dustan Levenstein
    Apr 20, 2015 at 3:44
  • $\begingroup$ It means it is injective, is that what you are going for? $\endgroup$
    – Math Student
    Apr 20, 2015 at 3:49
  • $\begingroup$ If the kernel is trivial, then the map is injective. I assume that's what you meant to say, and yes. $\endgroup$
    – Dustan Levenstein
    Apr 20, 2015 at 3:50
  • $\begingroup$ But I want the kernel to be nontrivial, so you are saying that what I said above is not true? $\endgroup$
    – Math Student
    Apr 20, 2015 at 3:50

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