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What are the usual tricks is proving that a group is not simple? (Perhaps a link to a list?)

Also, I may well be being stupid, but why if the number of Sylow p groups $n_p=1$ then we have a normal subgroup?

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    For your second question, you should notice that two subgroups are conjugate only if they are of the same order. So if I were to take a Sylow $p$-subgroup, $P$ say, and conjugate it by an element, $g\in G$, then $g^{-1}Pg$ is a subgroup of $G$ and it has the same order as $P$. As Sylow $p$-subgroups are defined by their order, if there is only one Sylow $p$-subgroup then $g^{-1}Pg=P$ for all $g\in G$. Thus, $P\lhd G$.2012-01-25
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    Thanks, @user1729, I have forgotten about that!2012-01-25

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