I am currently stuck on an exercise that asks to determine explicitly the subgroup $O_2(S_n)<S_n$ given by the intersection of all Sylow $2$-subgroups, for $n\geq 4$. I already proved that $O_2(S_n)$ is the unique largest normal $p$-subgroup of $S_n$, but I do not see how to use this to solve the exercise. Can somebody please give me a hint?
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1Well, do you know of any normal subgroup of $S_n$? – Mariano Suárez-Álvarez Jul 03 '15 at 23:33
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I don't understand what you mean by "$O_2(S_n)$ is the unique largest normal $p$-subgroup of $S_n$". Can you please explain this statement? – caffeinemachine Jul 04 '15 at 06:02