Rm×p

Rm×p denotes the direct product of the m-dimensional real vector space R^m with the cyclic group of order p, often written as Z_p or C_p. Its elements are pairs (x, k) where x ∈ R^m and k ∈ {0, 1, ..., p−1}. The group operation is defined componentwise: (x, k) · (y, ℓ) = (x + y, k + ℓ mod p). The identity element is (0, 0) and the inverse of (x, k) is (−x, −k).

Topologically, Rm×p carries the product topology, making it a Lie group that is a disjoint union of p copies of R^m, one for each k ∈ Z_p. Consequently, the group has m-dimensional smooth structure, and there are p connected components (each diffeomorphic to R^m). The identity component is R^m × {0}. The fundamental group is trivial, since each component is simply connected, while the set of connected components corresponds to Z_p.

In representation theory and harmonic analysis, Rm×p is a basic example of an abelian Lie group obtained by adjoining a finite discrete factor to a Euclidean space. Its Pontryagin dual is isomorphic to R^m × Z_p, reflecting the duality between the continuous and finite components. The group is often used to illustrate products of continuous and finite structures, as well as to model spaces in crystallography and other areas where a Euclidean space is combined with a finite symmetry.

Notes: Rm×p should not be confused with matrix spaces like R^{m×p} (the set of m by p real matrices) or with tensor products. The notation emphasizes a direct product with a finite cyclic group, not a space of matrices.