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A theorem proven by Frobenius states that

If $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$.

Articles discussing this theorem say that this result was proven by Frobenius in 1895, a precise reference given is

F. G. Frobenius, Verallgemeinerung des Sylow'schen Satzes, Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1895), 981-993

I am currently studying this theorem and its generalizations, and I would be interested in reading the original paper by Frobenius. However, I haven't been able to find a copy of it anywhere. Does anyone know where I could find it (preferably online)?

If the paper cannot be found, I would like to know what the original proof of Frobenius was like. The usual double induction proof can be found in Burnside's Theory of Finite Groups (1897), I guess the proof there might be very similar to the original proof of Frobenius. The same proof is also given in the book Theory and Applications of Finite Groups by G. A. Miller, H. F. Blichfeldt and L. E. Dickson (1916).

Also, according to Finkelstein [*], Frobenius discusses the theorem and its generalizations in the following papers:

  • F. G. Frobenius, Über auflösbare Gruppen, Sitzungberichte der Königl. Preuß. Akad. Wissenschaften (Berlin) (1893), 337-345.

  • F. G. Frobenius, Über endliche Gruppen,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1895), 81-112.

  • F. G. Frobenius, Verallgemeinerung des Sylow'schen Satzes,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1895), 981-993.

  • F. G. Frobenius, Über auflösbare Gruppen II,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1895), 1027-1044.

  • F. G. Frobenius, Über auflösbare Gruppen III,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1901), 849-875.

  • F. G. Frobenius, Über einen Fundamentalsatz der Gruppentheorie,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1903), 987-991.

  • F. G. Frobenius, Über einen Fundamentalsatz der Gruppentheorie II,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1907), 428-437.

Any idea where these could be found? Again, I haven't been able to find a copy of any of these papers and I'm not sure if they have been published anywhere afterwards.

[*] H. Finkelstein, Solving equations in groups: A survey of Frobenius' theorem Periodica Mathematica Hungarica Volume 9, Issue 3, pp 187-204, (1978).

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2 Answers 2

Here's the first paper you requested: Verallgemeinerung des Sylow'schen Satzes.

The paper begins on pdf page 459 of 695 (and on page 981 of the uploaded book).

I hope your German is okay!

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Thanks, but Google books does not allow me to see the paper for some reason.. –  Mikko Korhonen Oct 27 '12 at 18:12
    
That's very strange; the full paper is visible to me through google books. I'll download the pdf and then put it up on dropbox; tell me if you can see it. –  Benjamin Dickman Oct 27 '12 at 18:19
1  
I got it, thanks a lot for the help!! –  Mikko Korhonen Oct 27 '12 at 18:29
    
Excellent. Once you've slogged through this one, let me know if another paper is necessary and I'll try to root it out. –  Benjamin Dickman Oct 27 '12 at 21:23
up vote 1 down vote accepted

Some of these articles seem to be on Google Books, but I am unable to access them due to country restrictions. However, I was finally able to find the rest from Internet Archive.

All of these articles I was asking for were published in the journal "Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin". Searching with this keyword at the archive(click here) finds many (most?) issues of the journal published around 1890-1910, and fortunately I was able to find the articles I was looking for.

If you know some German, the first paper I was asking for is an interesting read. In the paper Frobenius proves his theorem and applies it to prove (for the first time) the following well known generalization of Sylow's theorem:

Let $G$ be a finite group and $p$ any prime. If $|G|$ is divisible by $p^k$, then the number of subgroups of order $p^k$ in $G$ is $\equiv 1 \mod{p}$.

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