# Cauchy's theorem or Frobenius' lemma

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