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I came up with following questions, while reading Wielandt's paper "Ein Beweis für die Existenz der Sylowgruppen". (I know the ideas of the proof, but my questions are related to some statements or comments in the original paper.) He starts with

In nicht wenigen Lehrbüchern der Algebra fehlt der Satz von SYLOW [= In not a few textbooks on Algebra, the Theorem of Sylow is missing (absent)].

Q.1 This paper is published in 1959. So, is it true that the Theorem of Sylow was absent in many algebra texts? (i.e. what are the popular algebra texts (before 1959) which do not mention this theorem?)

Q.2 In the proof, he says, let $|G|=p^{\alpha}r$. Let $\mathfrak{K}_1,\cdots, \mathfrak{K}_N$ be subsets (complexes) of $G$ of size $p^a$. I didn't understand the following statement.

Die Anzahl $N$ dieser „Komplexe“ $\mathfrak{K}_i$ ist nicht durch $p^{\varrho+1}$ teilbar, wenn $p^{\varrho}$ die höchste in $r$ aufgehende Potenz von $p$ bezeichnet;

Q.3 Finally, I couldn't understand his last suggestion:

Ähnlich ist schon Sylow selbst 1872 vorgegangen; eine kurze Durchführung findet man im Lehrbuch der Gruppentheorie von ZASSENHAUS, 1937, S. 100--101.

It would be grateful if one helps me to understand these statements.

In Q.1, I think "Algebra texts" means "abstract algebra texts", but not the school algebra; in other words, the texts would be introducing Groups, Subgroups, Lagranges theorem; but missing Sylow theorem.

azimut
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Groups
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  • In Q.3 I think he's referring you to the well known book by Zassenhaus to read about the "the same thing Sylow himself" did in 1872, or something like this. – Timbuc Dec 02 '14 at 06:01
  • Yes! I have seen the book of Zassenhaus, but what he want to say through "Sylow- 1872 - Zassenhaus", I couldn't understand. – Groups Dec 02 '14 at 06:06
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    I think it says something close to "Like the procedure (or method) of Sylow himself in 1872: a small application of it can be found in the text book in group theory by Zassenhauss, 1937, pages 100-101" . My german is great as long as no german speaker meets it, so the above might not be that accurate, but the general idea is that. – Timbuc Dec 02 '14 at 10:59
  • This is more clear to me. Here the book of Zassenhaus, 1937, is English edition of his book, or it is German? – Groups Dec 02 '14 at 11:02
  • Just to clarify Q.2, we have $r=p^\rho s$ where $p$ does not divide $s$, and the claim is that $N=\binom{p^{\alpha+\rho}s}{p^\alpha}$ is not divisible by $p^{\rho+1}$, which is correct but not obvious. You can use the Higman trick (see here for example) to avoid this step in the proof. – Derek Holt Feb 03 '23 at 09:09
  • Can everybody read German in Q.2 and Q.3? – citadel Feb 03 '23 at 12:32

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