Let $S=\langle g_1 , g_2, \cdots , g_l \rangle $ be a subgroup of a $n-$fold Pauli group $P_n$ generated by $g_1, g_2, \cdots, g_l$. Exercise 10.34 in Nielsen & Chuang Quantum Computation and Quantum Information requires to show that $S$ does not contain $-I$ if and only if $g_i ^2 = I $ and $g_i \ne -I$ for all $i=1,2,\cdots,l$.
The 'only if' part is not difficult, but I have a trouble with the 'if' part. How can I show that $S=\langle g_1 , g_2, \cdots , g_l \rangle $ does not contain $-I$ if $g_i ^2 = I $ and $g_i \ne -I$ for all $i=1,2,\cdots,l$ ?
I was wondering the same thing right now, it actually sounded false to me, so I looked for a counterexample, and I found one in the simplest Pauli group $P_1$!!
Consider the subgroup generated by the Pauli matrices $X=\begin{pmatrix} 0&1\\1&0\end{pmatrix}$ and $Z=\begin{pmatrix} 1&0\\0&-1\end{pmatrix}$. Clearly $X^2=Id$ and $Z^2=Id$. It is obvious that neither $X$ nor $Z$ are equal to $-Id$.
However, $XZXZ=-Id$.
This was the example I made, but the one mentioned in the comments by s.harp is even more obvious (s.harp takes $Z$ and $-Z$ as the generators of the subgroup).
Anyway, checking the rest of the section I don't think they use this result for anything, so it is not that relevant :) The "only if" is the important part.