Frucht is known for Frucht's theorem, the result that every group can be realized as the group of symmetries of an undirected graph, and for the Frucht graph, one of the two smallest cubic graphs without any nontrivial symmetries.
32.
The long-term effects of the Erlangen program can be seen all over pure mathematics ( see tacit use at congruence ( geometry ), for example ); and the idea of transformations and of synthesis using groups of symmetry has become standard in physics.
33.
This subgroup is isomorphic to the rotation group SO ( 3 ), which is the group of symmetries of the unit sphere in "'R "'3 ( which, when restricted to the sphere, become the isometries of the sphere ).
34.
A model geometry is called "'maximal "'if " G " is maximal among groups acting smoothly and transitively on " X " with compact stabilizers, i . e . if it is the maximal group of symmetries.
35.
The hemicube is the quotient { 4, 3 } / " N ", where " N " is a group of symmetries ( automorphisms ) of the cube with just two elements-the identity, and the symmetry that maps each corner ( or edge or face ) to its opposite.
36.
Yet this version of the stereographic image has a group of 4 symmetries in "'R "'3 ( it is isomorphic to the Klein 4-group ), as compared with the bounded version illustrated above having its group of symmetries the unique group of order 2 . ( If all symmetries and not just orientation-preserving isometries of "'R "'3 are allowed, the numbers of symmetries in each case doubles .)
37.
Note that the ADE correspondence is " not " the correspondence of Platonic solids to their reflection group of symmetries : for instance, in the ADE correspondence the tetrahedron, cube / octahedron, and dodecahedron / icosahedron correspond to E _ 6, E _ 7, E _ 8, while the reflection groups of the tetrahedron, cube / octahedron, and dodecahedron / icosahedron are instead representations of the Coxeter groups A _ 3, BC _ 3, and H _ 3.
38.
Every orientation-preserving automorphism of "'P "'1 ( 7 ) arises in this way, and so " G " = PSL ( 2, 7 ) can be thought of geometrically as a group of symmetries of the projective line "'P "'1 ( 7 ); the full group of possibly orientation-reversing projective linear automorphisms is instead the order 2 extension PGL ( 2, 7 ), and the group of collineations of the projective line is the complete symmetric group of the points.
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