Klein’s Erlangen Program: Groups Define Geometry

Klein’s Erlangen program: groups and geometry

There is a very deep link between group theory and geometry. Sadly, this link is not emphasized in most group-theory courses, even though the ideas are not difficult. The link was spelled out by Felix Klein in his Erlangen program; the goal of this article is to explain the basic ideas of that program.

Multiple geometries within Euclidean space

Start with familiar 2-dimensional Euclidean geometry. This is the geometry taught in high school, but it is worth noting there are actually several geometries hiding inside the same ambient set. Which geometries? The key point is that a “geometry” is determined by which transformations are declared to preserve “sameness” of geometric figures.

Consider triangles to illustrate the idea. Two triangles can be considered “the same” in different senses. Congruent triangles can be cut out of paper and placed on top of each other; they have equal angles and equal side lengths. Similar triangles have the same shape but possibly different sizes: all angles agree, while side lengths are proportional rather than equal.

Geometry as a relation defined by allowed transformations

In a geometry one specifies a relation on subsets of the underlying set: whether two geometric figures are regarded as “the same.” That relation is encoded by allowable transformations. For example, two triangles are congruent if there exists a transformation T: $\mathbb{R}^2\rightarrow \mathbb{R}^2$ that moves one triangle to the other. Such transformations include reflections, rotations, translations, or combinations of those. For similarity we also allow maps that uniformly shrink or expand the space.

Formally, given a set $X$ we equip it with a collection of allowable transformations. We call a subset $A$ of $X$ isomorphic to a subset $B$ if there is an allowable transformation $T$ with $T(A)=B$. Which transformations should be allowed? The choice is guided by natural desiderata:

• A set $A$ must be isomorphic to itself. This follows by including the identity map $T:X\rightarrow X$ among the allowable transformations.
• If $A$ is isomorphic to $B$ then $B$ should be isomorphic to $A$. To ensure this, allowable transformations should be invertible and the inverse of an allowable transformation should itself be allowable.
• If $A$ is isomorphic to $B$ and $B$ is isomorphic to $C$, then $A$ should be isomorphic to $C$. This is ensured by requiring that the composition of two allowable transformations is again allowable.

These three conditions mean the allowable transformations form a group. Thus a geometry can be defined as a set $X$ equipped with a group of transformations $X\rightarrow X$. Often we demand the group to act transitively on $X$; that is, for any $x,y\in X$ there exists a transformation $T$ with $T(x)=y$.

Examples of common geometries

• Euclidean geometry. The underlying set is $\mathbb{R}^2$ and the group is the Euclidean motions: bijections $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ that preserve distances. The Mazur–Ulam theorem implies these bijections are affine, and one obtains the familiar form $T(\mathbf{x}) = A\mathbf{x}+\mathbf{b}$ with $A\in O(2,\mathbb{R})$. Group-theoretically, the Euclidean motion group is the semidirect product of $O(2,\mathbb{R})$ with the translation group $\mathbb{R}^2$.
• Similarity geometry. Allow shrinking and expanding as well as rotations and translations. Transformations then have the form $T(\mathbf{x}) = \alpha A\mathbf{x} + \mathbf{b}$ with $A\in O(2,\mathbb{R})$ and $\alpha\neq 0$. Using complex numbers, the underlying set can be taken as $mathbb{C}$ and transformations written as $T(z)=az+b$ (see the linked complex-number discussion).
• Affine geometry. Allow all bijections that send straight lines to straight lines. Such maps preserve parallelism, and the fundamental theorem of affine geometry shows that these bijections are affine transformations. One can write them as $T(\mathbf{x}) = A\mathbf{x}+\mathbf{b}$ with $A\in GL(n,\mathbb{R})$.
• Other geometries. Hyperbolic and spherical geometry can likewise be described by appropriate transformation groups (Mobius transformations are relevant), and finite symmetry groups of solids (cubes, pyramids) fit the same scheme. For example, the symmetry group of a cube acting on its vertices is $S_4\times\mathbb{Z}_2$, with $48$ elements.

Invariants and moving to coarser geometries

Different geometries on the same set have different invariants — quantities preserved by the allowed transformations. In Euclidean geometry invariants include length, angles, and perpendicularity. In similarity geometry length is no longer invariant but angles remain invariant. In affine geometry angles are lost but parallel lines and ratios of lengths along a line remain invariant. At the extreme, if we equip $\mathbb{R}^2$ with the full group of homeomorphisms, we get topology: very few invariants survive, but dimension remains meaningful. Topology is sometimes called the largest interesting geometry.

Groups, homogeneous spaces and the pair (G,H)

One can also build geometries by specifying only a group and a subgroup. Given a group $G$ and a subgroup $H$, form the set of left cosets $X=G/H$. Each $g\in G$ defines a transformation $T_g: X\to X$ by $T_g(aH)=gaH$, so $G$ acts transitively on $X$. The subgroup $H$ is the isotropy group of the coset $H$ (the stabilizer of a chosen basepoint).

Conversely, if a group $G$ acts transitively on a geometry $X$, choose a point $p\in X$ and let $H=\{T\in G\mid T(p)=p\}$. Then $X$ is naturally isomorphic to $G/H$. Thus the geometric information is encoded in the pair $(G,H)$, with $G$ a group and $H$ a subgroup. For Euclidean geometry in two dimensions, the motion group is $E(2)$ and the stabilizer is $O(2)$, so $\mathbb{R}^2\cong E(2)/O(2)$. Many other examples fit the same pattern (for instance, a triangle as a geometry on three points can be described by a suitable pair like $(S_3, mathbb{Z}_2)$).

Extensions and further directions

From this perspective one can compare geometries by comparing their groups, or allow curvature and other structures to obtain Cartan–Cayley–Klein type geometries. The Erlangen program is a unifying viewpoint: a geometry is determined by its group of allowable transformations, and many classical geometries become homogeneous spaces $G/H$ under this lens.