I'm currently stuck on an exercise problem from Joseph Gallian's book, "Contemporary Abstract Algebra." The question is from Chapter 2, Exercise 12. It says: "For n>2, show that there are at least two elements in U(n) that satisfy $\ x^2=1$"
Where U(n) is the set of positive integers less than n and co-prime with n. This set is a group under multiplication mod n.
I see that a good way of showing this property is by induction. So I've set up my inductive hypothesis, after a couple of base cases of course (starting at n=3). But I'm stuck and I can't go from the hypothesis to the inductive step.
Any help would be appreciated. Thanks.
*Edit: Thanks everyone, I got it.