Conjugation of isometries in Euclidean space
Encyclopedia
In a group
, the conjugate by g of h is ghg−1.
Thus the conjugacy class
within the Euclidean group
E(n) of a translation is the set of all translations by the same distance.
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 containing a translation.
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:
Thus there is a partition
of the Euclidean group with in each subset one isometry that keeps the origin fixed, and its combination with all translations.
Each isometry is given by an orthogonal matrix
A in O(n) and a vector b:
and each subset in the quotient group is given by the matrix A only.
Similarly, for the special orthogonal group SO(n) we have
Thus the conjugacy class within the Euclidean group E(n) of inversion in a point is the set of inversions in all points.
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 (Rn).
Similarly { I, −I } is a normal subgroup
of O(n), and we have:
For odd n we also have:
and hence not only
but also:
For even n we have:
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.
The conjugate closure of a singleton containing a rotation in 3D is E+(3).
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.
E(2) has quotient group O(2) / Ck and E+(2) has quotient group SO(2) / Ck . For k = 2 this was already covered above.
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
, 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
are preserved.
Zmn / Zm
Zn.
Zmn is the direct product
of Zm and Zn if and only if m and n are coprime
. Thus e.g. Z12 is the direct product of Z3 and Z4, but not of Z6 and Z2.
This group, and more generally, abstract group Dihn, has the normal subgroup Zm for all divisors m of n, including n itself.
Additionally, Dih2n has two normal subgroups isomorphic with Dihn. They both contain the same group elements forming the group Zn, but each has additionally one of the two conjugacy classes of Dih2n \ Z2n.
In fact:
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, the conjugate by g of h is ghg−1.
Translation
If h is a translation, then its conjugate by an isometry can be described as applying the isometry to the translation:- the conjugate of a translation by a translation is the first translation
- the conjugate of a translation by a rotation is a translation by a rotated translation vector
- the conjugate of a translation by a reflection is a translation by a reflected translation vector
Thus the conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
within the Euclidean group
Euclidean group
In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
E(n) of a translation is the set of all translations by the same distance.
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
Conjugate closure
In group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by SG, i.e. the closure of SG under the group operation, where SG is the conjugates of the elements of S:The conjugate closure of S is denoted or G.The conjugate closure of any subset S of a group G...
of a singleton containing a translation.
Thus E(n) is a semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
O(n) and the subgroup of translations T, and O(n) is isomorphic with the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of E(n) by T:
- O(n)
E(n) / T
Thus there is a partition
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
of the Euclidean group with in each subset one isometry that keeps the origin fixed, and its combination with all translations.
Each isometry is given by an orthogonal matrix
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
A in O(n) and a vector b:
and each subset in the quotient group is given by the matrix A only.
Similarly, for the special orthogonal group SO(n) we have
- SO(n)
E+(n) / T
Inversion
The conjugate of the inversion in a point by a translation is the inversion in the translated point, etc.Thus the conjugacy class within the Euclidean group E(n) of inversion in a point is the set of inversions in all points.
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
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both 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.See also: Dihedral symmetry in three...
dih (Rn).
Similarly { I, −I } is a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of O(n), and we have:
- E(n) / dih (Rn)
O(n) / { I, −I }
For odd n we also have:
- O(n)
SO(n) × { I, −I }
and hence not only
- O(n) / SO(n)
{ I, −I }
but also:
- O(n) / { I, −I }
SO(n)
For even n we have:
- E+(n) / dih (Rn)
SO(n) / { I, −I }
Rotation
In 3D, the conjugate by a translation of a rotation about an axis is the corresponding rotation about the translated axis, etc.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.
The conjugate closure of a singleton containing a rotation in 3D is E+(3).
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.
E(2) has quotient group O(2) / Ck and E+(2) has quotient group SO(2) / Ck . For k = 2 this was already covered above.
Reflection
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).Rotoreflection
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 orientationIsometry groups
Two isometry groups are said to be equal up to conjugacy with respect to affine transformationAffine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
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
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s of two patterns which are both of a particular wallpaper group
Wallpaper group
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
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
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
, change of shape of the parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...
. 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
Orientation (mathematics)
In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
are preserved.
Cyclic groups
Cyclic groups are Abelian, so the conjugate by every element of every element is the latter.Zmn / Zm
Zn.Zmn is the direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of Zm and Zn if and only if m and n are coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
. Thus e.g. Z12 is the direct product of Z3 and Z4, but not of Z6 and Z2.
Dihedral groups
Consider the 2D isometry point group Dn. 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.This group, and more generally, abstract group Dihn, has the normal subgroup Zm for all divisors m of n, including n itself.
Additionally, Dih2n has two normal subgroups isomorphic with Dihn. They both contain the same group elements forming the group Zn, but each has additionally one of the two conjugacy classes of Dih2n \ Z2n.
In fact:
- Dihmn / Zn
Dihn - Dih2n / Dihn
Z2 - Dih4n+2
Dih2n+1 × Z2