Ray Tayek asked:

...

hi, iirc, the elements are : e, r, r2, r3, h, v, d1, and d2.

does h flip about the x or y axis?

is d1 / or \?

or are there a more mnemonic set of element names?

Those names seem pretty mnemonic to me -- I presume that your 'h' and 'v' flip over the 'horizontal' and 'vertical' axes. Of course, it doesn't matter which diagonal is called d1 and which is d2, it's all up to the person deciding to name them! To be as clear as possible, you could always name them id, rot(90), rot(180), rot(270), refl(/), refl(\), refl(-), refl(|). I think that's completely unambiguous. Or I did, until Eugene Salamin wrote:

The dihedral groups contain only proper rotations, no reflections or inversions. D4 has 8 elements. It has the 4-element subgroup C4 of rotations about the z-axis, and four 180-degree rotations about axes that lie in the xy-plane and are separated by 45 degrees.

Let's try to be clear here. "The dihedral group of order 8" is an abstract group of 8 elements, one of only two nonabelian such. If you're trying to picture it as actions, then you're doing representation theory, and Ray and Gene are disagreeing because they're thinking about two different representations. There's an action on the plane, in which (besides the identity) there are three rotations and four reflections; this is what Ray had in mind and where the names I listed above come from. There's also an action of the same group on 3-dimensional space, in which you only need rotations: things in the xy plane work just as in two dimensions (so the above is a subrepresentation), and anything that looked like a reflection also takes the z-axis to its negative, so it becomes a product of two reflections, which is a rotation. If you take a physical square and look at all of its symmetries, this is what you're doing: there are the obvious rotations, and four different ways to turn it over -- flip it horizontally, vertically, or along one of its diagonals. Of course, by "flip" here I don't mean a reflection, which is hard to accomplish with a physical object, but a 180-degree rotation which exchanges its top and its bottom. To a representation theorist, the 3-dim rotation rep is "decomposable" as a sum of the 2-dim rep on the fixed xy plane plus a 1-dim rep on the z-axis, taking the rotations to +1 and the reflections to -1. --Michael Kleber kleber@brandeis.edu