§ McKay's proof of Cauchy's theorem for groups [TODO ]
- In a group, if then . Prove this by writing .
- We can interpret this as follows: in the multiplication table of a group, firstly, each row contains exactly one .
- Also, when (ie, we are off the main diagonal of the multiplication table), each has a "cyclic permutation solution" .
- If the group as even order, then there are even number of s on the main diagonal.
- Thus, the number of solutions to for is even, since each solution has another paired with it.
- Let's generalize from pairs to