Suppose $G$ is a group and $H$ a subgroup of prime index $p$. I am trying to show that there exists an element $g\in G$ such that $$G/H=\{H,gH,g^2H,\ldots,g^{p-1}H\}$$ My attempt: By considering the (transitive) action of $G$ on $G/H$, we have a homomorphism $$\psi:G\longrightarrow S_p$$ and thus $\psi(G)$ is a transitive subgroup of $S_p$. It follows that $p$ divides $\psi(G)$ and thus, by Cauchy's theorem, there is an element $\psi(g)\in\psi(G)$ of order $p$. Then, $\psi(g^p)=\psi(g)^p=1$, so $g^p \in\ker\psi\subseteq H$, whence $g^pH=H$. The result would then follow if we can show that $g^k\notin H$ for $1\leq k\leq p-1$, but I didn't find a way to show that.
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1Look at Normal subgroup of prime index – leo Oct 20 '13 at 21:15
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@leo Here $p$ is not necessarily the smallest prime dividing $|G|$. How is that question useful for my question? – Spenser Oct 20 '13 at 21:21
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If $p$ were the minimal prime dividing $|G|$, then the linked post shows that the quotient $G/H$ is in fact a cyclic group, therefore such a $g$ must exist. – leo Oct 20 '13 at 21:24
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I thought you missed that assumption. Thanks for clarify that. – leo Oct 20 '13 at 21:27
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@leo Yes thanks. The question is indeed more general than that. $G/H$ might be just a set, not a group. – Spenser Oct 20 '13 at 21:28
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Why $\psi(G)$ is a transitive subgroup of $S_p$? and why $p$ divides $\psi(G)$? – eraldcoil May 25 '20 at 08:22
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1@eraldcoil Apply the orbit-stabilizer theorem. – caffeinemachine May 30 '20 at 05:56
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@caffeinemachine thanks – eraldcoil May 30 '20 at 06:45
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With the element $\psi(g)$ of order $p$, let us first verify that $g \notin H$. Suppose you had $g\in H$. Then $\psi(g^k)$ would fix $H$ for all $k$, so $\psi(g)$ would be a permutation of the $p-1$ other cosets of $H$, of order $p$. But $p \nmid (p-1)!$, so that can't be.
Hence we have $\psi(g)H \neq H$. Now consider the order of the orbit of $H$ under $\psi(g)^k$. Let that order be $n$. We saw $n > 1$. But $\psi(g)^p = \operatorname{id}$, hence $n \mid p$, so $n = p$, since $p$ is prime.
Daniel Fischer
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