in any geometric space have symmetry groups which are subgroups of the symmetries of the ambient space. Another example of a symmetry group is that of a combinatorial graph: a graph symmetry is a permutation of the vertices which takes edges to edges. Any finitely presented group is the symmetry group of its Cayley graph; the free group is the symmetry group of an infinite tree graph. Group structure in terms of symmetries Cayley's theorem states that any abstract group is a subgroup of the permutations of some set X, and so can be considered as the symmetry group of X with some extra structure. In addition, many abstract features of the group (defined purely in terms of the group operation) can be interpreted in terms of symmetries. For example, let G = Sym(X) be the finite symmetry group of a figure X in a Euclidean space, and let H ⊂ G be a subgroup. Then H can be interpreted as the symmetry group of X+, a "decorated" version of X. Such a decoration may be constructed as follows. Add some patterns such as arrows or colors to X so as to break all symmetry, obtaining a figure X# with Sym(X#) = {1}, the trivial subgroup; that is, gX# ≠ X# for all non-trivial g ∈ G. Now we get: Normal subgroups may also be characterized in this framework. The symmetry group of the translation gX + is the conjugate subgroup gHg−1. Thus H is normal whenever: that is, whenever the decoration of X+ may be drawn in any orientation, with respect to any side or feature of X, and still yield the same symmetry group gHg−1 = H. As an example, consider the dihedral group G = D3 = Sym(X), where X is an equilateral triangle. We may decorate this with an arrow on one edge, obtaining an asymmetric figure X#. Letting τ ∈ G be the reflection of the arrowed edge, the composite figure X+ = X# ∪ τX# has a bidirectional arrow on that edge, and its symmetry group is H = {1, τ}. This subgroup is not normal, since gX+ may have the bi-arrow on a different edge, giving a different reflection symmetry group. However, letting H = {1, ρ, ρ2} ⊂ D3 be the cyclic subgroup generated by a rotation, the decorated figure X+ consists of a 3-cycle of arrows with consistent orientation. Then H