ABSTRACT

In Chapter 2 we saw that Lagrange’s theorem has the consequence that if g G G and 0(0) = m, then m divides |G|. This raises the converse question: “If m divides |G| then does 0(0) = m for suitable g in G?” Our goal will be to prove this for m prime; for m not prime the result is false, as evidenced by Euler(8) = {1,3 ,5, 7}, which has order 4 although each of its elements has order 2. In proving Lagrange’s theorem we examined the process of division. In studying the converse we shall learn how to count.