The infinite dihedral group is .
Its irreducible complex representations have dimension at most two; the two-dimensional family pairs eigenvalues and of the rotation generator.
Every nonfaithful finite-dimensional complex representation of the infinite dihedral group is a direct sum of subrepresentations of dimension at most two. Its kernel contains a nonzero power of the rotation, so the representation factors through a finite dihedral group. Maschke's theorem makes that quotient representation semisimple, and all irreducible complex representations of a finite dihedral group have dimension at most two.
Let the rotation and reflection generators act on polynomials of degree at most two byThen and . The operator has one size-three unipotent Jordan block, so this three-dimensional representation cannot split into invariant subspaces of dimension at most two.
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The infinite dihedral group, usually denoted as \( D_{\infty} \) or sometimes \( D_{\infty}^* \), is a mathematical structure in group theory that extends the concept of the dihedral groups. While the finite dihedral group \( D_n \) represents the symmetries of a regular polygon with \( n \) sides (including rotations and reflections), the infinite dihedral group captures symmetries of an infinite linear arrangement.