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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.
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.
K 4: the Klein four-group of order 4, same as Z 2 × Z 2 and Dih 2; D 2n: the dihedral group of order 2n, the same as Dih n (notation used in section List of small non-abelian groups) S n: the symmetric group of degree n, containing the n! permutations of n elements
A two-dimensional representation of the Klein bottle immersed in three-dimensional space. In mathematics, the Klein bottle (/ ˈ k l aɪ n /) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.
D 2 is isomorphic to K 4, the Klein four-group. D 1 and D 2 are exceptional in that: D 1 and D 2 are the only abelian dihedral groups. Otherwise, D n is non-abelian. D n is a subgroup of the symmetric group S n for n ≥ 3. Since 2n > n! for n = 1 or n = 2, for these values, D n is too large to be a subgroup.
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
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.
The Klein four-group as a subgroup of S 4. See: Subgroups of S 4. There is also: left action last ⋅ first: This is: ... Dimensions User Comment; current: 23:32, 4 ...
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