# What are various approaches of proving that a permutation group is a subgroup of another?

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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The most obvious way is to show $g_1,g_2 \in G_2$. (Does this count as "trivial"?) –  Douglas S. Stones Oct 29 '12 at 7:29
OK, but how we can show that $g_1,g_2 \in G_2$? We should give a mapping from $\Gamma_1$ to $\Gamma_2$. But what if it's not possible? One can do the same for vector spaces over $\Gamma_i s$ –  Kaave Oct 29 '12 at 8:02
@Kaave: Are you thinking of $\Gamma$'s as vector spaces or just some sets? –  Babak S. Oct 29 '12 at 9:30
$\Gamma 's$ are just sets. –  Kaave Oct 29 '12 at 11:28
If the $\,\Gamma'\,$s are just sets then it suffices to show $\,\Gamma_1\subset \Gamma_2\,$ –  DonAntonio Oct 29 '12 at 11:57