cylindergroup

Cylindergroup is a term used in geometry to denote the symmetry group of a standard right circular cylinder, typically the surface S^1 × R in three-dimensional Euclidean space. As a set of isometries under composition, it encodes all rigid motions that preserve the cylindrical shape.

In its usual formulation, the Cylindergroup is the full isometry group of the cylinder surface. This group comprises rotations about the cylinder axis, translations along the axis, and reflections across planes that contain the axis. From a Lie group perspective, the orientation-preserving part of the group is isomorphic to SO(2) × R, which is topologically a product of a circle and a line. Including orientation-reversing symmetries expands the group to a structure isomorphic to O(2) × R, where O(2) accounts for the rotational and reflective symmetries of the cross-section and R accounts for axial translations. The full group thus has two connected components: the orientation-preserving component and the orientation-reversing component.

Key properties include its non-compactness and its status as a two-dimensional Lie group (the continuous part SO(2) × R has dimension 2; the discrete component from reflections does not add to the dimension). The maximal compact subgroup is SO(2), corresponding to pure rotations around the axis. The center of the continuous part is {I, -I} in O(2), and in the full group the center is isomorphic to {±I} × R.

Related topics include studies of cylindrical or helical symmetry in geometry, discrete subgroups yielding cylinder lattices, and applications in physics and engineering where cylindrical symmetry simplifies modeling.