Qualitative: A Uniformly Charged Cylinder

Homework Statement

This is just a qualitative question that is along side my main lecture with Griffiths EM book.

Basically we have a very long cylinder with charge density sigma and radius a along the z axis. This is mostly beside the point and is just to setup the question. The question asks that we are in a situation where a student thinks there are components of the E-field in every direction (for reference we use (s, phi, z) for the coordinates) outside the cylinder. It then asks you to explain each component of why you can or can't have them in, essentially, laymen terms (which I assume to be likely symmetry arguments). The s and phi I think I can do but the z I am having trouble discerning how symmetry says we can't have a z component.

The Attempt at a Solution

To start I know the electric field is only supposed to depend on the s component. However, I am confused about why we can't put an E-field on top of the cylinder and it thus would have components in this direction. I should mention that the cylinder is supposed to be very long which I am not sure if that means we don't care about the caps. In which case then if we put a test charge outside the cylinder then we move the cylinder up and down the test charge will not be affected because it looks exactly the same. However, lets say it is of length L how would this affect the answer exactly?