# Cauchy's theorem or Frobenius' lemma

A textbook exercises asks:

The goal of this exercise is to prove Frobenius's lemma, which asserts that if the order of the group $G$ is divisible by the prime $p$, then $G$ contains an element of order $p$.

The proof sketch given in the exercise is to consider the set of all ordered $p$-tuples of elements in $G$ whose components have a product equal to identity, and to show that a cyclic permutation that acts on the set of these $p$-tuples has at least $p$ fixed points. (I believe this proof is originally due to McKay, 1959).

I can solve this exercise, but my question here is just of historical curiosity - have you heard of this lemma before as being called Frobenius' lemma? It's usually only called Cauchy's theorem.

• What textbook is that? – DonAntonio Dec 23 '13 at 4:11
• @DonAntonio: This is from Godsil and Royle's Algebraic Graph Theory text, Exercise 2.5. – AG. Dec 23 '13 at 9:30

## 1 Answer

It seems that part of the history can be found in the Wikipedia page on Burnside's lemma. Burnside's lemma is also called the Cauchy-Frobenius lemma or the orbit-counting theorem. This relates the number of orbits of a group action to the cardinal of the stabilizers. This is a basic result in the theory of group actions, as the orbit-stabilizer theorem.

According to Wikipedia, Burnside attributed this lemma to an article of Frobenius of 1887, in his book "On the theory of groups of finite order", published in 1897. But it seems that this result was well-known at this time and appears in a work of Cauchy in 1845 and Burnside's lemma is also known as the "lemma that is not Burnside's".

Concerning the exact result you are talking about, I've always heard of Cauchy's lemma or theorem. I wouldn't be surprised that Cauchy proved it in its work of 1845 (but what is it ?) and I imagine that there has been some confusion, related to Burnside's lemma, since in all proofs of Cauchy's theorem that I know, Burnside's lemma is used in some way.

The complete works of Cauchy can be found in http://portail.mathdoc.fr/cgi-bin/oetoc?id=OE_CAUCHY_1_9. I guess that Cauchy's theorem appears in "Mémoire sur diverses propriétés remarquables des substitutions régulières ou irrégulières, et des systèmes de substitutions conjuguées" but it would take some time to find where since the terminology is very different from the today one. Just remember that they don't speak of groups but of systems of substitutions...

• I suppose if many of the proofs of Cauchy's theorem use Burnside's lemma in some way, that could have led to some confusion. The proof outlined in the exercise above doesn't require counting the number of orbits. – AG. Dec 23 '13 at 18:12