I have been reading the paper "[On the images of modular and geometric three-dimensional Galois representations][1]" (see also the earlier <a href="https://front.math.ucdavis.edu/0111.5043">arXiv preprint</a>). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.


The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.


We have an exact sequence, $1 \to R \to P(G) \to U \to 1$, where $R$ is diagonal and $U$ is a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to  $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.


I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic.
I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to <i> \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $<i>$, for example $<j>$, and $<k>$.


  [1]: MR2045504 11F80
Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.