5

In the paper "On A Theorem of Frobenius" written in 1969, Prof. Richard Brauer, for the first time, presented a character-free proof to the Frobenius theorem (i.e. counting the number of solutions to $g^n = e$ (more generally a conjugacy class of $G$)) without using double induction.

One of the most essential techniques being used is the following lemma (I have modified the notations used in the original paper for better reading):

(Brauer's Lemma):
Let $G$ be any group, and $N \triangleleft G$ a finite normal subgroup. Then for any $g \in G$ and $n \in N$ we have $(gn)^{|N|} \sim g^{|N|}$ (where "$|\cdot|$" denotes the cardinality, and "$\sim$" denotes the conjugation relation).

This lemma seems to be quite fundamental and useful. In Brauer's proof of the Frobenius theorem, it is the key to partitioning the solutions of $g^n = e$ into specially-designed equivalence classes with cardinalities of multiples of $n$. Moreover, Brauer himself gave one more application of his lemma at the end of the paper.

Nevertheless, I could not find any other references for this lemma. But I still think this result is quite interesting in itself.

Therefore, my questions are:

  1. Does anyone know any other results related to this lemma? What interesting consequences could this lemma imply?
  2. Does anyone have another proof for this lemma other than Brauer's? (Any method is welcome, need not be confined to "pure-group theory".)
  3. Are there any other fundamental results concerning conjugation relations?

Thank you very much for your help!

YCor
  • 60,149
Topoiii
  • 51
  • The paper may be found here: https://www.jstor.org/stable/2316779?seq=4#metadata_info_tab_contents – Topoiii Feb 19 '21 at 03:48
  • Related: https://mathoverflow.net/questions/109027/applications-of-frobenius-theorem-and-conjecture – Topoiii Feb 19 '21 at 03:50
  • 3
    "...Mr. Richard Brauer...." That would be Professor Richard Brauer, my mathematical grandfather. – Gerry Myerson Feb 19 '21 at 12:08
  • I am sorry for that. I have changed the wording. – Topoiii Feb 19 '21 at 12:39
  • 2
    The equivalence relations defined by Brauer come up in a paper of Isaacs: "Systems of equations and generalized characters in groups, Canad. J. Math., 22, 1040-1046, (1970)". – Mikko Korhonen Feb 19 '21 at 12:44
  • @Mikko Korhonen Thank you very much for the reference! – Topoiii Feb 19 '21 at 12:56
  • About $2.$, maybe there could a proof using Hall's formula $x^n y^n = (xy)^n c_2^{e_2} \cdots c_n^{e_n}$. Here $e_r = \binom{n}{r}$ and $c_r$ is in the $r$th term of the lower central series of $\langle x,y \rangle$. – spin Feb 20 '21 at 12:45
  • 1
    We used Brauer's lemma, see https://arxiv.org/abs/1806.08870 and https://arxiv.org/abs/2012.03123 . The first cited paper contains also a proof of this lemma, but as we wrote: ``We follow the original proof from [Bra69] but translate it into a more convenient (in our view) language.'' – Anton Klyachko Feb 22 '21 at 09:36
  • @Anton Klyachko Thank you very much for your papers on related theorems or properties! They are really interesting! Besides, I also realized that it is better to present Brauer's lemma using the "action" (but it is essentially the same as the original proof given by Prof. Brauer). – Topoiii Feb 22 '21 at 10:49

0 Answers0