It is often said that we can think of groups as the symmetries of some mathematical object. Usual examples involve geometrical objects, for instance we can think of $\mathbb{S}_3$ as the collection of all reflections and rotation symmetries of an equilateral triangle, similarly we can think of $D_8$ as the symmetry group of a square.
Cayley's Theorem along with the fact that the symmetry group of a regular $n$-simplex is isomorphic to $\mathbb{S}_{n+1}$ allows us to think of any finite group as a subset of the symmetry group of some geometrical object. Which brings me to the following questions:
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Can every finite group be represented as the collection of all symmetries of a geometrical object? That is, are all finite groups isomorphic to some Symmetry group?
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Can such a result (the representation of groups as distance-preserving transformations of some geometrical object) be extended to infinite groups? If so, how?
Thanks in advance (:
Best Answer
Yes. To any group $G$ (and choice of generating set $S$) you can associate its Cayley graph, which has a vertex for each group element $g$, and an edge between the vertices corresponding to $g$ and $gs$ for each $s$ in $S$. The left action of $G$ on itself corresponds to rigid motions of the graph. This graph is finite if and only if $G$ is a finite group.
If you know a little more topology, a corollary of Van Kampen's theorem is that every group $G$ is the fundamental group of a 2-dimensional CW complex $X$, so in particular the group $G$ acts by deck transformations on the universal cover $\tilde X$. It even turns out that every finitely presented group $G$ is the fundamental group of a 4-dimensional topological manifold. In the same vein, Eilenberg and Mac Lane gave a "functorial" construction of a (typically huge) geometric object $BG$, an example of what they term a $K(G,1)$—a space whose topology is in some sense completely determined by $G$, its fundamental group. This allows one to use methods from algebraic topology on even finite groups.
ETA: The representation of infinite, discrete groups as distance-preserving transformations of geometric objects is a central concern of Geometric Group Theory! Meier's Groups, Graphs and Trees or Clay and Margalit's Office Hours With a Geometric Group Theorist make excellent introductions to this field.