Operations
What a symmetry actually is
Not a property of a shape but a motion that leaves it alone. Once symmetry is a verb rather than an adjective, everything else in the subject follows — including why there can only ever be seventeen wallpapers.
The four motions of the plane
Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.
Why it is a group and not a list
The symmetries of a pattern cannot be chosen independently. Do two of them in succession and the result is forced to be a third, which is why there is no eighteenth wallpaper for anybody to invent.
The orbit is the pattern
A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out.