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In mathematics, D 3 (sometimes alternatively denoted by D 6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S 3. It is also the smallest non-abelian group. [1] This page illustrates many group concepts using this group as example.
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.
Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. [3] The notation for the dihedral group differs in geometry and abstract algebra. In geometry, D n or Dih n refers to the symmetries of the n-gon, a group of order 2n.
One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a ...
The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. [3] Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6. [4]
S 3 (dihedral group of order 6) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements, and the latter is how it is realized by its regular representation. *
The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and inspection of the symmetry of their Cayley tables verifies this. In contrast, the smallest non-abelian group, the dihedral group of order 6, does not have a symmetric Cayley table.
Its subgroup of rotations is the dihedral group D n of order 2n, which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. Note: in 2D, D n includes reflections, which can also be viewed as flipping over flat objects without distinction of frontside and backside; but in 3D, the two operations are ...