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