In a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
, the conjugate by ''g'' of ''h'' is ''ghg''
−1.
Translation
If ''h'' is a translation, then its conjugation by an isometry can be described as applying the isometry to the translation:
*the conjugation of a translation by a translation is the first translation
*the conjugation of a translation by a rotation is a translation by a rotated translation vector
*the conjugation of a translation by a reflection is a translation by a reflected translation vector
Thus the
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
within the
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformati ...
''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. So, this is the
conjugate closure
In group theory, the normal closure of a subset S of a group G is the smallest normal subgroup of G containing S.
Properties and description
Formally, if G is a group and S is a subset of G, the normal closure \operatorname_G(S) of S is the ...
of a
singleton containing a translation.
Thus ''E''(''n'') is a
direct product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
of the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
''O''(''n'') and the subgroup of translations ''T'', and ''O''(''n'') is isomorphic with the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
of ''E''(''n'') by ''T'':
:''O''(''n'')
''E''(''n'') ''/ T''
Thus there is a
partition of the Euclidean group with in each subset one isometries that keeps the origins fixed, and its combination with all translations.
Each isometry is given by an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identi ...
''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
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
dih (''R''
''n'').
Similarly is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of ''O''(''n''), and we have:
:''E''(''n'') ''/'' dih (''R''
''n'')
''O''(''n'') ''/''
For odd ''n'' we also have:
:''O''(''n'')
''SO''(''n'') ×
and hence not only
:''O''(''n'') ''/'' ''SO''(''n'')
but also:
:''O''(''n'') ''/''
''SO''(''n'')
For even ''n'' we have:
:''E''
+(''n'') ''/'' dih (''R''
''n'')
''SO''(''n'') ''/''
Rotation
In 3D, the conjugate by a translation of a rotation about an axis is the corresponding rotation about the translated axis. Such a conjugation produces the
screw displacement
In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a screw displacement. A direct Euclidean isometry in three dimensions involves a translation and a rotation. The ...
known to express an arbitrary Euclidean motion according to
Chasles' theorem.
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) ''/ C
k'' and ''E''
+(2) has quotient group ''SO''(2) ''/ C
k'' . 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 orientation
Isometry groups
Two isometry groups are said to be equal up to conjugacy with respect to
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More general ...
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
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
s of two patterns which are both of a particular
wallpaper group
A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry, symmetries in the pattern. Such patterns occur frequently in architecture a ...
type. If we would just consider conjugacy with respect to isometries, we would not allow for scaling, and in the case of a parallelogrammatic
lattice, change of shape of the
parallelogram
In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
. 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 may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
are preserved.
Cyclic groups
Cyclic groups are Abelian, so the conjugate by every element of every element is the latter.
''Z''
''mn'' ''/ Z''
''m'' ''Z''
''n''.
''Z''
''mn'' is the
direct product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
of ''Z''
''m'' and ''Z''
''n'' if and only if ''m'' and ''n'' are
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
. Thus e.g. ''Z''
12 is the direct product of ''Z''
3 and ''Z''
4, but not of ''Z''
6 and ''Z''
2.
Dihedral groups
Consider the 2D isometry point group ''D''
''n''. 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 Dih
''n'', has the normal subgroup Z
''m'' for all divisors ''m'' of ''n'', including ''n'' itself.
Additionally, Dih
2''n'' has two normal subgroups isomorphic with Dih
''n''. They both contain the same group elements forming the group Z
''n'', but each has additionally one of the two conjugacy classes of Dih
2''n'' \ ''Z''
2''n''.
In fact:
:Dih
''mn'' / ''Z
n''
Dih
''n''
:Dih
2''n'' / Dih
''n'' ''Z''
2
:Dih
4''n''+2 Dih
2''n''+1 × ''Z''
2
References
{{DEFAULTSORT:Conjugation Of Isometries In Euclidean Space
Euclidean symmetries
Group theory