An element in a camera lens refers to a single piece of optical glass or plastic. The group in a lens description refer to either single elements, or two or more elements that are in contact over the whole of their surfaces. So a lens with four elements in contact in one lump and two elements that don't touch any other has six elements in three groups. This web page has some lens diagrams which might make this clear:
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All straight lines on a photo taken with a rectilinear wide angle lens will be shown as straight. This ensures that photos of buildings and other things with straight lines do not appear distorted. With a fisheye lens, only straight lines that go through the exact centre of the photo will be shown as straight. All others will be curved.
Now consider a rectilinear, very wide angle lens set up to take a photo of a building. The camera is set up so that the sensor is parallel to the face of the building. The middle of the building is perhaps 30 feet in front of, and also 30 feet away from the lens, but the ends of the building, still 30 feet in front of the lens, are (diagonally) 42 feet away. Yet the photo shows the ends of the building to the same scale as the middle; they do not diminish in size due to perspective.
There's a snag. Although buildings look alright, three dimensional, roughly spherical objects that we are familiar with do not. Here I mean people's faces and bodies. Look at this page, and scroll down to the 'Complement your wide-angle lens' section at the bottom. Click on the before and after buttons below the wedding photo:
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It's most noticeable on faces but can make people look very fat. The further from the centre, the worse the effect. Any lens wider than about 19mm on a Pentax DSLR will show this effect at the edges and especially the corners to an unflattering degree.
Fisheye lenses don't seem to me to show this effect to anywhere near the same degree, but I'm still experimenting with my lens. If you change lenses from the very wideangle lens in the building shot above to a fisheye lens, the ends of the building will look smaller in line with the laws of perspective. Any straight lines in the building will be curved, unless they go though the exact centre of the lens.
Scenes without any clues to distortion like buildings or people in them can be taken with a fisheye lens without the fisheye effect being noticeable.
Rectilinear very wideangle lenses usually have a very small, almost unnoticeable amount of barrel distortion; it's very difficult to get rid of it altogether. Barrel distortion is what fisheye lenses have in abundance.