Geometry & Topology 2024, 32: Geometry & Group Theory

Geometry can be discussed in terms of transformation groups. For a space X and group G made up of transformations of X, the geometric notions are quantities measured on X which are invariant under the action of G. This can be formalized rigorously by way of connecting group transformations to the geometric notion of symmetry. A transformation of a set X is a bijective map T: X → X, so then transformations can be composited without issue. In fact, all compositions of transformations in X will result in an already existing transformation in X. The notions of a neutral element and the inverse element for any particular transformation are thus included. A transformation group is a subgroup of the full set of transformations of X. Subgroups - for a reminder - contain the Identity element and are closed under operation (composition) and the inverse. It "retains the group structure". So far, Eucl(n) ⊂ Aff(n) ⊂ PGL(n+1).

For a set X, and g, T transformations of X. If T has some properties expressed in terms of data from X. The conjugate transformation gTg⁻¹ is determined by the same properties in terms of g applied to the same data. Aff(n) has 2 distinct subgroups: the translation subgroup, which is isomorphic to ℝⁿ and the subgroup GL(n)₀ of linear maps, which is isomorphic to GL(n). Aff(n) is the direct product of these subgroups. A normal subgroup H of a group G is taken to itself by conjugacy in G. This means the translation subgroup ℝⁿ ⊂ Aff(n) is a normal subgroup. GL(n)₀ is a subgroup of Aff(n) and is not normal. The first projection (A, b) → A defines a surjective group homomorphism mapping isomorphically to GL(n). The kernel of Aff(n) → GL(n) is ℝⁿ, and the action of GL(n) on ℝⁿ is conjugacy in Aff(n). In Euclidean space, is a reflection group.