I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, “How do you represent a group geometrically in a space?” Is there any way of representing it?
This is a natural question; the short answer is (1) yes, and (2) that this can be an instructive and powerful way to understand particular groups. In fact, this perspective is so natural, that modern students are sometimes surprised that groups were not invented for this purpose. (Rather, Galois introduced them to study what are now called Galois groups, that is, the groups of automorphisms of splitting fields of polynomials, which is an almost entirely symbolic, rather than geometric, enterprise.)
Narrowing our scope, for any group G, we can ask whether there is some subset X of Rn such that the group of symmetries of X (more precisely, the group of isometries of Rn that preserve X as a set) is isomorphic to G. This is the case for several familiar groups:
- S2 is the isometry group of a line segement
- S3, equilateral triangle
- S4, regular tetrahedron
- (more generally) the symmetric group Sn, regular n-simplex, which for concreteness we can take to be the convex hull of the points (0,…,0,1,0,…,0) in Rn.
- the Klein 4-group Z2×Z2, (nonsquare) rectangle
- D8, square
- D10, regular pentagon
- the dihedral group D2n, regular n-gon
We can produce more familiar examples by imposing additional conditions on the symmetries, e.g., by requiring that they preserve the orientation of the set X:
- A3≅Z3, oriented symmetries of the equilateral triangle, or just the symmetries of a triskelion
- Z4, a square (oriented)
- Z5, a regular pentagon (oriented)
- the cyclic group Zn, a regular n-gon (oriented)
- A4, a regular tetrahedron (oriented)
- the alternating group An, a regular n-simplex (oriented)
- S4, a cube (or octahedron) (oriented)
- A5, a dodecahedron (or icosahedron) (oriented) (this one in particular is perhaps not so easy to see immediately: given a dodecahedron, one can draw five distinguished cubes inside it, and each [oriented] symmetry of the dodecahedron permutes these in a unique alternating way, that is, A5 is the alternating group on the set of these cubes).
One can also ask about groups with infinitely many elements:
- SO(2)≅S1 is the group of oriented symmetries of the circle S1, which we can also think of as the group of oriented linear transformations of R2 preserving the Euclidean inner product
- the special orthogonal group SO(n) is the group of oriented symmetries of the n-sphere, which we can also think of as the group of oriented linear transformations of Rn+1 preserving the Euclidean inner product
If we expand our scope to permit more exotic geometries, we can find new classes of examples, for examples, projective planes over finite fields:
- GL(3,2)≅PGL(3,2)=PSL(3,2), the group of automorphisms of the Fano plane P(F32), that is, the projective plane over the finite field F2 of two elements (this group has 168 elements, and after A5, is the second smallest finite simple group of nonprime order). It is nonobvious that this group is “accidentally” isomorphic to PSL(2,7), the group of automorphisms of the projective line P(F27) over the field F7 of seven elements.
Generally the projective special linear groups PSL(n,pk) are unfamiliar to a beginner, but there are a few exceptions that give us new ways to view familiar groups:
One can expand on these lists (which should be regarded only as collections of examples, and not in any way exhaustive) wildly by generalizing in various ways what exactly one means by geometric.
Aside Surely this answer is already long enough, but I’ll point out that the converse to your question is natural and important, too: For any geometric object X, we can ask for the group G of symmetries of X. This too is a deep font of interesting examples, but I’ll mention just a few related families of examples, the first two of which have tractible classifications and the third of which has a famous application:
- If X is a pattern in R2 that repeats “infinitely, in one direction”, the symmetry group of X is one of the 7 frieze groups; one of these is Z∞≅Z, and the rest are variations on Z and an infinite analogue D∞:=Z⋊Z2 of Dn.
- If X is a pattern in R2 that repeats “infinitely, in two directions”, one gets one of the 17 wallpaper groups. The simplest of these are Z×Z, Z×D∞, and D∞×D∞.
- Asking analogous questions about patterns in R3 leads to the study of space groups, of which there are hundreds, and some of which are of critical importance in chemistry because of their appearance in regular crystal structures.