I have a problem in a textbook that is giving me trouble. Either I'm missing something or there is some sort of mistake in the question. The problem is with part (ii), the other parts seem okay, but I'll post all parts anyway:

Let $\displaystyle p = 2^k +1$ be a prime number, and let $\displaystyle G$ be the group of integers $\displaystyle 1, 2, \ldots, p-1$, with multiplication modulo $\displaystyle p$.

(i) Show that if $\displaystyle 0 < m < k$ then $\displaystyle 0 < 2^m -1 < p$ and deduce that $\displaystyle 2^m \neq 1 \mod p$.

(ii) Show that if $\displaystyle k< m < 2k$ then $\displaystyle 2^m = 1 \mod p \Longrightarrow 2^{2k-m} = -1 \mod p$ and decude that $\displaystyle 2^m \neq 1 \mod p$

(iii) Use parts (i) and (ii) to show that the order of the element 2 in $\displaystyle G$ is $\displaystyle 2k$

(iv) Decude that $\displaystyle k$ is a power of 2.