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| {{Unreferenced|date=December 2009}}
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| In a [[Group (mathematics)|group]], the '''conjugate''' by ''g'' of ''h'' is ''ghg''<sup>−1</sup>.
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| ==Translation==
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| If ''h'' is a translation, then its conjugate by an isometry can be described as applying the isometry to the translation:
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| *the conjugate of a translation by a translation is the first translation
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| *the conjugate of a translation by a rotation is a translation by a rotated translation vector
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| *the conjugate of a translation by a reflection is a translation by a reflected translation vector
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| Thus the [[conjugacy class]] within the [[Euclidean group]] ''E''(''n'') of a translation is the set of all translations by the same distance.
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| The smallest subgroup of the Euclidean group containing all translations by a given distance is the set of ''all'' translations. Thus this is the [[conjugate closure]] of a [[singleton (mathematics)|singleton]] containing a translation.
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| Thus ''E''(''n'') is a [[semidirect product]] of the [[orthogonal group]] ''O''(''n'') and the subgroup of translations ''T'', and ''O''(''n'') is isomorphic with the [[quotient group]] of ''E''(''n'') by ''T'':
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| :''O''(''n'') <math>\cong</math> ''E''(''n'') ''/ T''
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| Thus there is a [[Partition of a set|partition]] of the Euclidean group with in each subset one isometry that keeps the origin fixed, and its combination with all translations.
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| Each isometry is given by an [[orthogonal matrix]] ''A'' in ''O''(''n'') and a vector ''b'':
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| :<math>x \mapsto Ax+ b</math>
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| and each subset in the quotient group is given by the matrix ''A'' only. | |
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| Similarly, for the special orthogonal group ''SO''(''n'') we have
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| :''SO''(''n'') <math>\cong</math> ''E''<sup>+</sup>(''n'') ''/ T''
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| ==Inversion==
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| The conjugate of the [[inversion in a point]] by a translation is the inversion in the translated point, etc.
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| Thus the conjugacy class within the Euclidean group ''E''(''n'') of inversion in a point is the set of inversions in all points.
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| Since a combination of two inversions is a translation, the conjugate closure of a singleton containing inversion in a point is the set of all translations and the inversions in all points. This is the generalized [[dihedral group]] dih (''R''<sup>''n''</sup>).
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| Similarly { ''I'', −''I'' } is a [[normal subgroup]] of ''O''(''n''), and we have:
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| :''E''(''n'') ''/'' dih (''R''<sup>''n''</sup>) <math>\cong</math> ''O''(''n'') ''/'' { ''I'', −''I'' }
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| For odd ''n'' we also have:
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| :''O''(''n'') <math>\cong</math> ''SO''(''n'') × { ''I'', −''I'' }
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| and hence not only
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| :''O''(''n'') ''/'' ''SO''(''n'') <math>\cong</math> { ''I'', −''I'' } | |
| but also:
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| :''O''(''n'') ''/'' { ''I'', −''I'' } <math>\cong</math> ''SO''(''n'')
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| For even ''n'' we have:
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| :''E''<sup>+</sup>(''n'') ''/'' dih (''R''<sup>''n''</sup>) <math>\cong</math> ''SO''(''n'') ''/'' { ''I'', −''I'' }
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| ==Rotation==
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| In 3D, the conjugate by a translation of a rotation about an axis is the corresponding rotation about the translated axis, etc.
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| Thus the conjugacy class within the Euclidean group ''E''(3) of a rotation about an axis is a rotation by the same angle about any axis.
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| The conjugate closure of a singleton containing a rotation in 3D is ''E''<sup>+</sup>(3).
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| In 2D it is different in the case of a ''k''-fold rotation: the conjugate closure contains ''k'' rotations (including the identity) combined with all translations.
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| ''E''(2) has quotient group ''O''(2) ''/ C<sub>k</sup>'' and ''E''<sup>+</sup>(2) has quotient group ''SO''(2) ''/ C<sub>k</sup>'' . For ''k'' = 2 this was already covered above.
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| ==Reflection==
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| The conjugates of a reflection are reflections with a translated, rotated, and reflected mirror plane. The conjugate closure of a singleton containing a reflection is the whole ''E''(''n'').
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| ==Rotoreflection==
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| The left and also the right coset of a reflection in a plane combined with a rotation by a given angle about a perpendicular axis is the set of all combinations of a reflection in the same or a parallel plane, combined with a rotation by the same angle about the same or a parallel axis, preserving orientation
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| ==Isometry groups==
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| Two isometry groups are said to be equal up to conjugacy with respect to [[affine transformation]]s if there is an affine transformation such that all elements of one group are obtained by taking the conjugates by that affine transformation of all elements of the other group. This applies for example for the [[symmetry group]]s of two patterns which are both of a particular [[wallpaper group]] type. If we would just consider conjugacy with respect to isometries, we would not allow for scaling, and in the case of a parallelogrammetic [[lattice (group)|lattice]], change of shape of the [[parallelogram]]. Note however that the conjugate with respect to an affine transformation of an isometry is in general not an isometry, although volume (in 2D: area) and [[orientation (mathematics)|orientation]] are preserved.
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| ==Cyclic groups==
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| Cyclic groups are Abelian, so the conjugate by every element of every element is the latter.
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| ''Z''<sub>''mn''</sub> ''/ Z''<sub>''m''</sub> <math>\cong</math> ''Z''<sub>''n''</sub>.
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| ''Z''<sub>''mn''</sub> is the [[direct product of groups|direct product]] of ''Z''<sub>''m''</sub> and ''Z''<sub>''n''</sub> if and only if ''m'' and ''n'' are [[coprime]]. Thus e.g. ''Z''<sub>12</sub> is the direct product of ''Z''<sub>3</sub> and ''Z''<sub>4</sub>, but not of ''Z''<sub>6</sub> and ''Z''<sub>2</sub>.
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| ==Dihedral groups==
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| Consider the 2D isometry point group ''D''<sub>''n''</sub>. The conjugates of a rotation are the same and the inverse rotation. The conjugates of a reflection are the reflections rotated by any multiple of the full rotation unit. For odd ''n'' these are all reflections, for even ''n'' half of them.
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| This group, and more generally, abstract group Dih<sub>''n''</sub>, has the normal subgroup Z<sub>''m''</sub> for all divisors ''m'' of ''n'', including ''n'' itself.
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| Additionally, Dih<sub>2''n''</sub> has two normal subgroups isomorphic with Dih<sub>''n''</sub>. They both contain the same group elements forming the group Z<sub>''n''</sub>, but each has additionally one of the two conjugacy classes of Dih<sub>2''n''</sub> \ ''Z''<sub>2''n''</sub>.
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| In fact:
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| :Dih<sub>''mn''</sub> / ''Z<sub>n''</sub> <math>\cong</math> Dih<sub>''n''</sub>
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| :Dih<sub>2''n''</sub> / Dih<sub>''n''</sub> <math>\cong</math> ''Z''<sub>2</sub>
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| :Dih<sub>4''n''+2</sub> <math>\cong</math> Dih<sub>2''n''+1</sub> × ''Z''<sub>2</sub>
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| {{DEFAULTSORT:Conjugation Of Isometries In Euclidean Space}}
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| [[Category:Euclidean symmetries]]
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| [[Category:Group theory]]
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Greetings! I am Myrtle Shroyer. North Dakota is exactly where me and my husband live. Managing individuals is what I do and the wage has been truly fulfilling. Body building is what my family and I enjoy.
Also visit my weblog: std testing at home - click the up coming post -