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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?

Travis Willse
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    Many groups have an inherent geometric interpretation, such as the dihedral groups, the symmetry groups of the archimedean solids and wallpaper groups. Also, see Cayley graph. – Arthur Apr 26 '15 at 15:05
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    You will definitely want to check out the book *Visual Group Theory* by Nathan Carter: http://books.google.com/books/about/Visual_Group_Theory.html?id=T_o0CnMZecMC – Ben Blum-Smith Apr 26 '15 at 16:59
  • I'd suggest you get hold of *Groups: A Path To Geometry* by R. P. Burn. It has a very distinctive didactic style, probably quite distinct from any other textbook in the area, even if the content is standard. – Silverfish Apr 27 '15 at 00:14

2 Answers2

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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 $\mathbb{R}^n$ such that the group of symmetries of $X$ (more precisely, the group of isometries of $\mathbb{R}^n$ that preserve $X$ as a set) is isomorphic to $G$. This is the case for several familiar groups:

  • $S_2$ is the isometry group of a line segement
  • $S_3$, equilateral triangle
  • $S_4$, regular tetrahedron
  • (more generally) the symmetric group $S_n$, regular $n$-simplex, which for concreteness we can take to be the convex hull of the points $(0, \ldots, 0, 1, 0, \ldots, 0)$ in $\mathbb{R}^n$.
  • the Klein $4$-group $Z_2 \times Z_2$, (nonsquare) rectangle
  • $D_8$, square
  • $D_{10}$, regular pentagon
  • the dihedral group $D_{2n}$, 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 $X$:

  • $A_3 \cong Z_3$, oriented symmetries of the equilateral triangle, or just the symmetries of a triskelion
  • $Z_4$, a square (oriented)
  • $Z_5$, a regular pentagon (oriented)
  • the cyclic group $Z_n$, a regular $n$-gon (oriented)
  • $A_4$, a regular tetrahedron (oriented)
  • the alternating group $A_n$, a regular $n$-simplex (oriented)
  • $S_4$, a cube (or octahedron) (oriented)
  • $A_5$, 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, $A_5$ is the alternating group on the set of these cubes).

Cube in a dodecahedron

One can also ask about groups with infinitely many elements:

  • $SO(2) \cong {\Bbb S}^1$ is the group of oriented symmetries of the circle ${\Bbb S}^1$, which we can also think of as the group of oriented linear transformations of $\mathbb{R}^2$ 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 $\mathbb{R}^{n + 1}$ 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:

  • $GL(3, 2) \cong PGL(3, 2) = PSL(3, 2)$, the group of automorphisms of the Fano plane $\Bbb P(\Bbb F_2^3)$, that is, the projective plane over the finite field $\Bbb F_2$ of two elements (this group has $168$ elements, and after $A_5$, 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 $\Bbb P (\Bbb F_7^2)$ over the field $\Bbb F_7$ of seven elements.

Generally the projective special linear groups $PSL(n, p^k)$ are unfamiliar to a beginner, but there are a few exceptions that give us new ways to view familiar groups:

  • $PSL(2, 2) \cong S_3$
  • $PSL(2, 3) \cong A_4$
  • $PSL(2, 4) \cong PSL(2, 5) \cong A_5$
  • $PSL(2, 9) \cong A_6$
  • $PSL(4, 2) \cong A_8$

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 $\mathbb{R}^2$ that repeats "infinitely, in one direction", the symmetry group of $X$ is one of the $7$ frieze groups; one of these is $Z_{\infty} \cong {\Bbb Z}$, and the rest are variations on $\Bbb Z$ and an infinite analogue $D_{\infty} := \Bbb Z \rtimes Z_2$ of $D_n$.
  • If $X$ is a pattern in $\mathbb{R}^2$ that repeats "infinitely, in two directions", one gets one of the $17$ wallpaper groups. The simplest of these are $\Bbb Z \times \Bbb Z$, ${\Bbb Z} \times D_{\infty}$, and $D_{\infty} \times D_{\infty}$.
  • Asking analogous questions about patterns in $\mathbb{R}^3$ 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.
Travis Willse
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    It's funny, because the historical development has been quite the converse, I think (admitting a meager education). The goal (since Klein, or Cartan?) has been to try to use potent algebraic constructs to understand and treat the classical geometries. Or even worse, "doing geometry" in sets that are *a priori* just groups (such as the ones just mentioned that were constructed to study... geometry). – GPerez Apr 26 '15 at 16:19
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    Is there some group which cant be represented in this way? –  Apr 26 '15 at 17:20
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    @Amudhan That's a good question, but certainly the answer depends on what you mean by "this way". I know of (even small, finite) groups for which I know of no satisfying geometric interpretation. On the other hand, every finite group can be regarded as a subset of $GL(N, \mathbb{R})$ for some $N$, so every finite group can be regarded as a collection of *some* symmetries of some vector space $\mathbb{R}^N$. – Travis Willse Apr 26 '15 at 17:28
  • @Travis so every group can be looked as a symmetry? –  Apr 27 '15 at 01:12
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    @Rememberme: This is effectively [Cayley's theorem](http://en.wikipedia.org/wiki/Cayley%27s_theorem). – Nate Eldredge Apr 27 '15 at 02:04
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    @Rememberme Every group can be regarded as a collection of symmetries of some $\mathbb{R}^N$, yes, but it's not always transparent what the characterization of those symmetries actually *is*, that is, what is the extra structure on that space that is preserved exactly by the group. – Travis Willse Apr 27 '15 at 02:19
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    Graph Theory supplies an answer. For every (finite) group $G$, there is a graph such that the group of symmetries of the graph is $G$. – Gerry Myerson Apr 27 '15 at 07:10
  • @GerryMyerson What is the name of this result? Is it constructive? – Travis Willse Apr 28 '15 at 01:20
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    @Travis: [Frucht's Theorem](http://en.wikipedia.org/wiki/Frucht%27s_theorem) "states that every finite group is the group of symmetries of a finite undirected graph". By the way: Once you have a graph, you can construct a (possibly-degenerate) geometric realization, for which each automorphism corresponds to a "rigid motion"; I describe those realizations (which I call "spectral") [in this answer](http://math.stackexchange.com/a/309001/409), where I also provide a link to a PDF with much greater detail. – Blue Apr 28 '15 at 01:36
  • There are also Cayley graphs - every group (finite or not) is the group of symmetries of a directed graph. – user1729 Apr 28 '15 at 09:39
  • @user1729 The digraph must also have labeled edges (according to the corresponding edges), correct? – Travis Willse Apr 28 '15 at 09:46
  • @Travis Sorry, I've deleted my previous comment as it was wrong. Yes, you are correct - the edges do need to be labelled. However, I suspect that there is a clever way to get around this. (For example, if your generating set is finitely generated then number the generators, and partition each edge corresponding to the $k$th generator into $k$ pieces. This probably won't work, but something cleverer might.) – user1729 Apr 28 '15 at 13:22
  • @user1729 I think one needs to modify your construction slightly, so that the subgraphs we use to replace the edges are asymmetric; otherwise, it's ambiguous whether a given edge encodes a generator or its inverse. Probably it's enough to partition each edge into three edges and "tag" the new node connected to (say) the head of each directed edge with $k$ loops, where $k$ denotes the index of the generator in some ordering. – Travis Willse Apr 28 '15 at 13:38
  • @Travis I was still going for a directed graph! Your method fails with cyclic groups, but I think would work if you labelled the start and end of the edges (say label the start with $2n$ loops, and end with $2n+1$ loops). Alternatively, in my construction label the start "mini-edge" with a single loop. I think both ideas probably work, but wouldn't bet on either! – user1729 Apr 28 '15 at 15:05
  • @user1729 I think there's another problem with my idea, namely that adding more than one loop to a node will introduce automorphisms that permute those loops. Anyway, I think it's more or less clear /some/ variation on the idea work, one simply needs to replace a directed edge with something with no internal symmetries (and that doesn't otherwise introduce some other new symmetry). This comment thread has gotten somewhat long, so if you want to discuss some aspect of this further, perhaps we should move to chat. – Travis Willse Apr 28 '15 at 15:10
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You might find interesting a computer app called Group Explorer.

The app provides visualizations of 59 groups. Additional groups are available as separate downloads.

Some screenshots:

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VividD
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