mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the...
27 KB (3,380 words) - 17:32, 4 August 2024
mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry...
6 KB (615 words) - 19:44, 23 August 2023
longitudinal dihedral angle of a fixed-wing aircraft Dihedral group, the group of symmetries of the n-sided polygon in abstract algebra Also Dihedral symmetry...
857 bytes (147 words) - 15:41, 30 October 2023
four-group is the smallest non-cyclic group. It is, however, an abelian group, and isomorphic to the dihedral group of order (cardinality) 4, symbolized...
10 KB (1,357 words) - 07:18, 20 August 2024
known as the binary dihedral group. The connection with the binary cyclic group C2n, the cyclic group Cn, and the dihedral group Dihn of order 2n is illustrated...
8 KB (1,229 words) - 14:10, 9 November 2023
wrote "a3b" as a shorthand for a3 ∘ b. In mathematics this group is known as the dihedral group of order 8, and is either denoted Dih4, D4 or D8, depending...
20 KB (1,789 words) - 11:12, 14 July 2024
cyclic group C4 and the Klein four-group V4 are both 2-groups of order 4, but they are not isomorphic. Nor need a p-group be abelian; the dihedral group Dih4...
21 KB (2,753 words) - 13:08, 25 October 2023
In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive...
5 KB (623 words) - 23:46, 13 December 2022
symmetry group. the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group Dih(R). Up...
17 KB (2,283 words) - 19:34, 22 March 2024
generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the...
6 KB (875 words) - 19:34, 19 March 2023
non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in...
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dihedral group 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...
18 KB (2,657 words) - 20:57, 19 July 2024
the dihedral group 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...
33 KB (1,313 words) - 04:08, 31 October 2023
Semidirect product (category Group products)
are non-abelian groups: the dihedral group of order 16 the quasidihedral group of order 16 the Iwasawa group of order 16 If a given group is a semidirect...
30 KB (4,541 words) - 06:25, 10 August 2024
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...
26 KB (3,724 words) - 08:50, 20 June 2024
This permutation group is known, as an abstract group, as the dihedral group of order 8. In the above example of the symmetry group of a square, the permutations...
23 KB (3,367 words) - 18:53, 18 June 2024
matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is...
101 KB (13,126 words) - 14:06, 5 August 2024
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...
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explains the name dihedral group. An object having symmetry group Cn, Cnh, Cnv or S2n has rotation group Cn. An object having symmetry group Dn, Dnh, or Dnd...
60 KB (5,094 words) - 00:10, 3 September 2024
S3 is the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries...
46 KB (6,130 words) - 06:34, 24 May 2024
G is an extension of a finite group by either the infinite cyclic group or the infinite dihedral group. "Cyclic group", Encyclopedia of Mathematics,...
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root of unity. The group Gal ( L / Q ) {\displaystyle \operatorname {Gal} (L/\mathbb {Q} )} is isomorphic to S3, the dihedral group of order 6, and L...
18 KB (3,190 words) - 20:36, 19 July 2024
Quotienting a group by its commutator subgroup Dihedral group of order 6 – Non-commutative group with 6 elements, the smallest non-abelian group Elementary...
36 KB (5,284 words) - 19:55, 26 August 2024
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...
46 KB (5,637 words) - 12:41, 27 June 2024
inversion in the origin: the group of all translations and inversion in all points; this is the generalized dihedral group of R3, Dih(R3). E(1), E(2),...
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In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it...
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orthogonal group Oε(2, q) is a dihedral group of order 2(q − ε), where ε = ±. Proof For studying the orthogonal group of Oε(2, q), one can suppose that...
56 KB (7,844 words) - 08:45, 30 June 2024
products will not uniquely describe an element of G. For example, the dihedral group D8 of order sixteen can be generated by a rotation, r, of order 8; and...
22 KB (2,428 words) - 08:01, 8 July 2024
In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...
64 KB (9,427 words) - 12:29, 18 August 2024
finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called...
11 KB (1,337 words) - 08:48, 12 July 2024