Another Proof of Cauchy's Group Theorem

Since ab = 1 implies ba = b(ab)b−1 = 1, the identities are symmetrically placed in the group table of a finite group. Each row of a group table contains exactly one identity and thus if the group has even order, there are an even number of identities on the main diagonal. Therefore, x = 1 has an even number of solutions. Generalizing this observation, we obtain a simple proof of Cauchy’s theorem. For another proof see [1].