Some Consequences of Cauchy’s Theorem

Abstract

In this handout, we meet some basic consequences of Cauchy's theorem in group theory. These consequences do not depend on the proof of Cauchy's theorem, but only on the conclusion of the theorem. Proof. If |G| is a power of p, then the order of any element of G is a power of p since the order of any element divides the size of the group. Conversely, assume all elements of G have p-power order. To show |G| is a power of p, suppose it is not, so |G| is divisible by a prime q = p. Then, by Cauchy, G has an element of order q, and that's a contradiction of our assumption. Theorem 1.2. If all non-identity elements of G have the same order, this order is a prime p and |G| is a power of p. Proof. If |G| has two prime factors, say p and q, then G contains elements of orders p and q by Cauchy, which contradicts the hypothesis. Thus |G| has only one prime factor, say |G| = p m for a prime p. The orders of a non-identity element could be {p, p 2 , . . . , p m }. However, by Cauchy some g ∈ G has order p, so the hypothesis tells us every non-identity element of G has order p. Example 1.3. Abelian groups fitting the hypothesis of Theorem 1.2 are easy to write down, e.g, (Z/(p)) n where p is any prime. Every non-zero element has order p. For a nonabelian example, consider the Heisenberg group over Z/(p) when p is an odd prime. Every non-identity element has order p. Can you find a nonabelian example when p = 2?