Any fixed-point symmetry of a lattice transforms lattice points into new lattice points.
Its matrix elements must therefore all be integers. As a consequence also the trace
trace = 2 cos(theta) + n-2
of any lattice rotation must be integer. n is the dimension of the space concerned. Together with the cyclic condition theta = 2Pi/p for a rotation generating a rotation subgroup we get:
p = 1,2,3,4,6
but not 5,7,8, etc. The applet below shows a two-dimensional lattice with a six fold rotation symmetry (p=6). You can drag the red line interactively to rotate the green lattice. The six corners of the central hexagon correspond to six symmetry rotations, which are elements of 2C_6. You can even drag the center of the green lattice to different positions. That way you can verify, that it doesn't matter around which center of one of the black hexagons you rotate. The rotation symmetry is for rotations around any hexagon center the same. Compare also applets for p=5 and p=7 illustrating why 5 and 7 fold two dimensional lattices don't exist.
Soli Deo Gloria. Created with Cinderella1.2 by Eckhard Hitzer (Fukui).