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I have two permutation groups $G_1=\langle g_1,g_2\rangle$ and $G_2=\langle h_1,h_2\rangle$ acting on $\Gamma_1$ and $\Gamma_2$ respectively. I want to prove that $G_1 \leq G_2$. What are more nontrivial methods to do so?

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    Technically the permutation group $G_1$ cannot be a subgroup of the permutation group $G_2$ unless $\Gamma_1=\Gamma_2$. If $\Gamma_1\subseteq\Gamma_2$ then $G_1$ can be isomorphic to a subgroup of $G_2$.2012-10-29

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