How to find a p-Sylow for symmetric group $S_{p^2}$？

Asked Nov 10 '19 at 07:28

Active Nov 10 '19 at 08:22

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I basically have no clue. It seems this $S_{p^2}$ symmetric group has $p^2!$ order which means the $p^n$ should be $p^{p+1}$. So it seems that the Sylow group should have this order. And then I have no idea about such an order's meaning.

Did I try the wrong path? The question also asked me to describe this group abstractly as a semi-direct product. But I simply stuck and can't find a specific one.

edited Nov 10 '19 at 08:06

asked Nov 10 '19 at 07:28

qwertymask

Seems it should be $p^{p+1}$. E.g. $p=3$, $9!$ has factors $3$, $6$, $9$ giving $3^4$. – RghtHndSd Nov 10 '19 at 07:49
Yeah, thank you for this... However, is there something special with p+1? – qwertymask Nov 10 '19 at 08:13
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The first duplicate target handles $S_{p^2}$. In the more general case of $S_{p^n}$ we get a tower of wreath products. An IMO useful description of that construction is given in Ted's answer here. – Jyrki Lahtonen Nov 10 '19 at 08:24

How to find a p-Sylow for symmetric group $S_{p^2}$？

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