I'm stuck on one part of a problem.

"suppose $\displaystyle \Phi(n)=2t$ where t is odd, and $\displaystyle n=p^e$ for odd prime p and positive integer e.

Let L be the subgroup of false witnesses, i.e. L is the set of congruence classes made up of integers b prime to n such that $\displaystyle b^{(n-1)/2)}\equiv \left(\frac{b}{n}\right)\ mod n$

Now suppose the order of L is $\displaystyle |L|=\Phi(n)/2=t$.

Let g be a primitive root mod n,so g has order $\displaystyle \Phi(n)$ and every element prime to n is of the form $\displaystyle g^j mod n$ for some j.

Show that for every integer $\displaystyle g^j$ belonging to a congruence class in L, j must be even. Since |L|=t, explain why $\displaystyle L=g^{2i}: i=1....t$"

I really don't know what to do here so any help would be appreciated.