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Algebraic structure → Group theory Group theory
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The symmetry group of a snowflake is D6, a dihedral symmetry, the same as for a regular hexagon.
In mathematics, a dihedral group is the group of symmetries of a regular polygon,[1][2] which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
The notation for the dihedral group differs in geometry and abstract algebra. In geometry, Dn or Dihn refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D2n refers to this same dihedral group.[3] This article uses the geometric convention, Dn.
^Weisstein, Eric W. "Dihedral Group". MathWorld.
^Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
^"Dihedral Groups: Notation". Math Images Project. Archived from the original on 2016-03-20. Retrieved 2016-06-11.
mathematics, a dihedralgroup is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedralgroups are among the...
mathematics, the infinite dihedralgroup Dih∞ is an infinite group with properties analogous to those of the finite dihedralgroups. In two-dimensional geometry...
longitudinal dihedral angle of a fixed-wing aircraft Dihedralgroup, the group of symmetries of the n-sided polygon in abstract algebra Also Dihedral symmetry...
known as the binary dihedralgroup. The connection with the binary cyclic group C2n, the cyclic group Cn, and the dihedralgroup Dihn of order 2n is illustrated...
symmetry group. the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedralgroup Dih(R). Up...
wrote "a3b" as a shorthand for a3 ∘ b. In mathematics this group is known as the dihedralgroup of order 8, and is either denoted Dih4, D4 or D8, depending...
finite groups, it can be realized as the Galois group of a certain field of algebraic numbers. The quaternion group Q8 has the same order as the dihedral group...
generalized dihedralgroups are a family of groups with algebraic structures similar to that of the dihedralgroups. They include the finite dihedralgroups, the...
dihedralgroup of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group. This page illustrates many group...
are non-abelian groups: the dihedralgroup of order 16 the quasidihedral group of order 16 the Iwasawa group of order 16 If a given group is a semidirect...
the dihedralgroup of order 2n (often the notation Dn or D2n is used) K4: the Klein four-group of order 4, same as Z2 × Z2 and Dih2 D2n: the dihedral group...
explains the name dihedralgroup. An object having symmetry group Cn, Cnh, Cnv or S2n has rotation group Cn. An object having symmetry group Dn, Dnh, or Dnd...
This permutation group is known, as an abstract group, as the dihedralgroup of order 8. In the above example of the symmetry group of a square, the permutations...
S3 is the first nonabelian symmetric group. This group is isomorphic to the dihedralgroup of order 6, the group of reflection and rotation symmetries...
geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as...
Quotienting a group by its commutator subgroup Dihedralgroup of order 6 – Non-commutative group with 6 elements, the smallest non-abelian group Elementary...
Z(C_{n})={\frac {1}{n}}\sum _{d|n}\varphi (d)a_{d}^{n/d}.} The dihedralgroup is like the cyclic group, but also includes reflections. In its natural action,...
an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations...
inversion in the origin: the group of all translations and inversion in all points; this is the generalized dihedralgroup of R3, Dih(R3). E(1), E(2),...
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time...
In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...
alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree...
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