# dihedral group

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## dihedral group

n.
The group of rotations and reflections of a regular polygon.
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[D.sub.n] = (a,b|[b.sup.2] = [a.sup.k] = 1, [b.sup.-1]ab = [a.sup.-1]) - the dihedral group of order n = 2k;
(i) The dihedral group [D.sub.8] in order 8: this group has 4 elements of order 4.
This is most transparent in the class of diagrams of highest symmetry, namely, Figure 1, and the diagrams symmetric under the n-th dihedral group, Figures 2, 3(a), and 4.
The language of [t.sub.b,m] is closed under a group isomorphic to the dihedral group of order 2m, here denoted [I.sub.2](m).
i) Any infinite tree without leaf is a building of type (W, S) where W is the infinite dihedral group Z/2Z * Z/2Z and S = {(1, 0), (0, 1)}.
Finally, we consider the Cambrian lattice Ck associated with the dihedral group [D.sub.k], see for instance .
We constructed a finite group and well know group dihedral group such that the corresponding non-commuting graphs are non isomorphic but the group have the same order.
According to group theory in mathematics, the dihedral group of order 3 is one of the most robust.
The Group Calculator allows the user to select a group from one of several families: cyclic group of order n, dihedral group [D.sub.n] of order 2n, the group [Z.sup.*.sub.n] of units modulo n, the abelian group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the affine group Aff([Z.sub.n]) = {ax + b | a [member of] [Z.sup.*.sub.n] and b [member of] [Z.sub.n]} under composition.
As examples we have [S.sub.3], [A.sub.4], the dihedral group [D.sub.2m], m odd.
The generators [rho] and [tau] of the dihedral group [D.sub.n] = {e,[rho],...,[[rho].sup.n- 1],[tau],...,[tau][[rho].sup.n-1]} are such that [[rho].sup.n] = e, [[tau].sup.2] = e and [tau][rho] = [[rho].sup.n- 1][tau].
It is the dihedral group [D.sub.n] of order 2n with I = (1) = (xb)[.sup.n] = [x.sup.2] = (b)[.sup.2].

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