This has as its automorphism groups F_4 whose root space is a minimal tessellation in 4 dimensions and G_2. Einstein Still Rules, Says Fermi Telescope Team | Universe Today

Sym (V) is also a model of set theory with set of atoms A, and Ï induces an automorphism of Sym (V). The Axiom of Choice

But since Ï is an automorphism, it also preserves function application, so that Ï (f (U)) = Ïf (Ï (U)). The Axiom of Choice

If is a model of Z (Zermelo set theory) with an external automorphism Ï and an ordinal Îº such that Îº Ï (Îº), then Quine's New Foundations

Inasmuch as a mirror reflection of the space is a transformation of the entire space into itself it is called an automorphism. SYMMETRY AND ASYMMETRY

If such a group is given and held fast, then a figure in space is called symmetric if each automorphism of the group leaves it unchanged. SYMMETRY AND ASYMMETRY

If the space is a so-called metric space, it is usually expected that the underlying automorphism shall at least be isometric, meaning that the transformations preserve the distance between any pair of points. SYMMETRY AND ASYMMETRY

Obviously the homogeneity of a space is the more interesting if the underlying group of automorphism is more important. SYMMETRY AND ASYMMETRY

In general, an automorphism of a space, any space, is an invertible transformation of the entire space into itself, and such a transformation is nothing other than a rearrangement of the totality of the points of the space from their given ordering into any other. SYMMETRY AND ASYMMETRY