electrostatics - Why is the electric field of an infinite insulated plane of charge perpendicular to the plane?

I'm studying Gauss' Law, and I came across a section where we're supposed to find the electric field of various shapes (like an infinite line of charges, etc), and for an infinite plane with a uniform positive charge per area, it says here in my notes:

Planar symmetry => Charge distribution doesn't change if we slide it in any direction parallel to the sheet => At each point, the field is perpendicular to the sheet, and it must have the same magnitude at any given distance on either side of the sheet.

It's not clear to me why having a charge distribution that doesn't change will result in a field perpendicular to the sheet. Can anyone help me clarify?

Answer

The answer by @NowIGetToLearnWhatAHeadIs is correct. It's worth learning the language used therein to help with your future studies. But as a primer, here's a simplified explanation.

Start with your charge distribution and a "guess" for the direction of the electric field.

As you can see, I made the guess have a component upward. We'll see shortly why this leads to a contradiction.

Now do a "symmetry operation," which is a fancy phrase for "do something that leaves something else unchanged. In this case, I'm going to reflect everything about a horizontal line. I mean everything.

The "top" of the sheet became the "bottom." This is just arbitrary labeling so you can tell I flipped the charge distribution. The electric field is flipped too. (Imagine looking at everything in a mirror, and you'll realize why things are flipped the way they are.)

Hopefully everything is okay so far. But now compare the original situation with the new inverted one.

You have exactly the same charge distribution. You can't tell that I flipped it, except for my arbitrary labeling. But if you have the same charge distribution, you ought to also have the same electric field. As you can see, this is not the case, which means I made a mistake somewhere.

The only direction for the electric field that does not lead to this contradiction is perpendicular to the sheet of charge.