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V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.
List of all nonabelian groups up to order 31 Order Id. [a] G o i Group Non-trivial proper subgroups [1] Cycle graph Properties 6 7 G 6 1: D 6 = S 3 = Z 3 ⋊ Z 2: Z 3, Z 2 (3) : Dihedral group, Dih 3, the smallest non-abelian group, symmetric group, smallest Frobenius group.
D 2, [2,2] +, (222) of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has three perpendicular 2-fold rotation axes. It is the symmetry group of a cuboid with an S written on two opposite faces, in the same orientation. D 2h, [2,2], (*222) of order 8 is the symmetry group of a cuboid.
The group G is called the principal group of the geometry and G/H is called the space of the geometry (or, by an abuse of terminology, simply the Klein geometry). The space X = G/H of a Klein geometry is a smooth manifold of dimension dim X = dim G − dim H. There is a natural smooth left action of G on X given by
A Guide to the Classification Theorem for Compact Surfaces is a textbook in topology, on the classification of two-dimensional surfaces. It was written by Jean Gallier and Dianna Xu , and published in 2013 by Springer-Verlag as volume 9 of their Geometry and Computing series ( doi : 10.1007/978-3-642-34364-3 , ISBN 978-3-642-34363-6 ).
L 2 (3) ≅ A 4 A 3 ≅ C 3 via the quotient by the Klein 4-group; L 2 (5) ≅ A 5. To construct such an isomorphism, one needs to consider the group L 2 (5) as a Galois group of a Galois cover a 5: X(5) → X(1) = P 1, where X(N) is a modular curve of level N. This cover is ramified at 12 points.
Any hyperbolic 3-manifold of finite volume has a finite cover whose fundamental group surjects onto a non-abelian free group (such groups are usually called large). Another conjecture (also proven by Agol) which implies 1-3 above but a priori has no relation to 4 is the following : 5.
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