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 $x^2=1$"
Here $U(n)$ 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've got it.