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Let $G=A^m \rtimes S_m$ where $A$ is some abelian group. Now what can I say about sylow $2$-subgroup of $G$. The text I am reading says let $S$ be the fixed sylow $2$-subgroup $S_2 \rtimes S(12)$. Which group is this, I did not get it and how do one fine sylow 2-subgroups of wreath products. I have not done any such theorem earlier. And what is $S_2 \rtimes S(12)$, if it is clear to you please tell me.

Thanks

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  • This question discusses Sylow $2$-subgroups of the symmetric group. They can, indeed, be described using iterated wreath products of $S_2$. That shows in both the answers there. Probably something better exists on our site as well. Understandably I found that question fast. – Jyrki Lahtonen Mar 15 '16 at 13:27
  • In general a Sylow subgroup of $A^m\rtimes S_m$ can be described by specifying Sylow $p$-subgroups $P_A$ and $P$ for $A$ and $S_m$ respectively, and then taking the subgroup $Q:=P_A^m\rtimes P.$ It has the correct order. Here $P_A^m$ is stable under the permutation action of all of $S_m$, so $Q$ is a subgroup. – Jyrki Lahtonen Mar 15 '16 at 13:32

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