Let $p$ be a prime. Prove that $(\mathbb{Z}_p\wr\mathbb{Z}_p)\wr \mathbb{Z}_p$ is isomorphic to a $p$-Sylow subgroup of $S_{p^3}$. Here, $\wr$ denotes the wreath product and $\mathbb{Z}_p$, the cyclic group of order $p$. Please explain in detail.
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3This is advanced-basic group theory: wreath product and stuff. You surely already did something. Please write down your own work on this. – DonAntonio Sep 20 '16 at 19:11
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1@DietrichBurde Can you give a good reference for its proof? – Manish Kumar Sep 20 '16 at 19:38
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1Yes, take Huppert's book on finite groups. – Dietrich Burde Sep 20 '16 at 19:39
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1IIRC this is (in full generality) a guided exercise in Jacobson's Basic Algebra I. See Ted's answer for a nice alternative description. The description in terms of automorphisms of a full binary tree for the 2-Sylow subgroups of $S_{2^n}$ is particularly nice! – Jyrki Lahtonen Sep 20 '16 at 19:43
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The structure of the Sylow $p$-subgroups of $S_n$ is indeed known: The Sylow $p$-subgroup of the symmetric group on $n=p^k$ letters is
$$ \underbrace{C_p \wr C_p \wr \cdots \wr C_p}_{k}, $$
that is, an $k$-fold wreath product of cyclic groups of order $p$. See Huppert, Endliche Gruppen I, Satz III.15.3 for a proof.
Dietrich Burde
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Huppert's Endliche Gruppen I, one of the most awesome books in group theory, and still there is not a traduction to english! I guess we got used to the german language and to the frightening gothic letters (much worse than the language itself...) – DonAntonio Sep 20 '16 at 20:16
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I still have books at home full of gothic letters...it took me a while to be able to read it. A bit of cyrillic is also awesome, like the Tate-Shafarevich group $Ш_{E/K}$. – Dietrich Burde Sep 20 '16 at 20:30