Symmetry group

In group theory, the symmetry group of a geometric object comprises all transformations under which the object remains invariant, with composition serving as the group operation. Each transformation is an invertible mapping of the ambient space that maps the object onto itself while preserving its pertinent structural attributes. The symmetry group of an object X is frequently represented by the notation G = Sym(X).

Within a metric space, the symmetries of an object constitute a subgroup of the ambient space's isometry group. While this discussion primarily addresses symmetry groups within Euclidean geometry, the underlying concept extends to a broader array of geometric structures.

Introduction

Objects exhibiting symmetry are defined as geometric figures, images, and patterns, exemplified by a wallpaper design. When considering the symmetry of physical entities, their material composition can also be incorporated into the pattern definition. (Formally, a pattern can be conceptualized as a scalar field—a position-dependent function yielding values such as colors or substances—a vector field, or a more generalized function defined over the object.) The group of spatial isometries induces a group action on embedded objects, wherein the symmetry group Sym(X) comprises those isometries that map X onto itself, including any associated patterns. An object X is deemed invariant under such a mapping, and the mapping itself is termed a symmetry of X.

The aforementioned concept is occasionally referred to as the full symmetry group of X, specifically to highlight its inclusion of orientation-reversing isometries (e.g., reflections, glide reflections, and improper rotations), provided these transformations map the object X onto itself. Conversely, the subgroup consisting solely of orientation-preserving symmetries (such as translations, rotations, and their combinations) is designated as the proper symmetry group. An object is characterized as chiral if it lacks any orientation-reversing symmetries, implying that its proper symmetry group is identical to its full symmetry group.

A symmetry group whose elements share a common fixed point—a condition met if the group is finite or the figure is bounded—can be represented as a subgroup of the orthogonal group O(n) by positioning the origin at this fixed point. In such cases, the proper symmetry group becomes a subgroup of the special orthogonal group SO(n) and is termed the rotation group of the figure.

Within a discrete symmetry group, points symmetric to any specified point do not converge towards a limit point. Consequently, each orbit of the group—defined as the set of images of a particular point under all group elements—constitutes a discrete set. It follows that all finite symmetry groups are inherently discrete.

Discrete symmetry groups are categorized into three distinct types: (1) finite point groups, which encompass only rotations, reflections, inversions, and rotoinversions, effectively representing the finite subgroups of O(n); (2) infinite lattice groups, characterized by containing solely translational elements; and (3) infinite space groups, which incorporate elements from both preceding categories, potentially alongside additional transformations such as screw displacements and glide reflections. Furthermore, continuous symmetry groups (Lie groups) exist, featuring rotations of infinitesimally small angles or translations over arbitrarily small distances. A notable illustration is O(3), which represents the symmetry group of a sphere. The symmetry groups of Euclidean objects can be exhaustively classified as subgroups of the Euclidean group E(n), which is the isometry group of Rⁿ.

Two geometric figures are considered to possess the same symmetry type if their respective symmetry groups are conjugate subgroups within the Euclidean group. This implies that subgroups H§67§ and H§1011§ are related by the expression H§1516§ = g⁻¹H§2324§g for some element g belonging to E(n). Illustrative examples include:

• For instance, two three-dimensional figures may exhibit mirror symmetry, albeit with respect to distinct mirror planes.
• Similarly, two three-dimensional figures might possess 3-fold rotational symmetry, yet around different axes.
• Furthermore, two two-dimensional patterns could display translational symmetry, each along a single direction, where the associated translation vectors share identical lengths but diverge in orientation.

Subsequent discussions focus exclusively on isometry groups whose orbits exhibit topological closure, encompassing both discrete and continuous isometry groups. This criterion, however, precludes groups such as the one-dimensional group of rational number translations. Figures defined by such non-closed orbits are inherently difficult to represent accurately, owing to their infinitely intricate detail.

One Dimension

In one dimension, the isometry groups include:

• The trivial cyclic group, denoted C₁.
• Groups consisting of two elements, generated by a single reflection. These groups are isomorphic to C₂.
• Infinite discrete groups formed by a translation operation. These are isomorphic to Z, which represents the additive group of integers.
• Infinite discrete groups generated by both a translation and a reflection. These groups are isomorphic to the generalized dihedral group of Z, specifically Dih(Z), also designated as D_∞ (a semidirect product of Z and C§89§).
• The group generated by all possible translations, which is isomorphic to the additive group of real numbers, R. This particular group cannot serve as the symmetry group for any Euclidean figure, even if patterned, because such a pattern would inherently be homogeneous and thus also susceptible to reflection. Nevertheless, a constant one-dimensional vector field does possess this symmetry group.
• The group formed by all translations combined with reflections at specific points. This group is isomorphic to the generalized dihedral group Dih(R).

Two Dimensions

Considering conjugacy, the discrete point groups in two-dimensional space are categorized into the following classes:

• Cyclic groups, denoted C₁, C₂, C§45§, C§6_{7§, and so forth, where Cn encompasses all rotations around a fixed point by angular increments that are multiples of 360°/n.}
• Dihedral groups, including D₁, D₂, D§45§, D§6_{7§, and subsequent groups, where Dn (with an order of 2n) comprises the rotations found in Cn, augmented by reflections across n axes that intersect the fixed point.}

C₁ represents the trivial group, containing only the identity operation, and is characteristic of asymmetric figures, such as the letter "F". C₂ corresponds to the symmetry group of the letter "Z", C§45§ to that of a triskelion, and C§6_{7§ to a swastika. Groups such as C§8}9§, C§10_{11§, and others denote the symmetry groups of analogous swastika-like figures possessing five, six, or more arms, respectively, rather than four.}

D₁ is a two-element group comprising the identity operation and a singular reflection. This group characterizes figures possessing only one axis of bilateral symmetry, exemplified by the letter "A".

D₂, isomorphic to the Klein four-group, functions as the symmetry group for a non-equilateral rectangle. This geometric figure exhibits four distinct symmetry operations: the identity, a single twofold axis of rotation, and two non-equivalent mirror planes.

D₃, D₄, and similar groups represent the symmetry groups of regular polygons.

For each of these symmetry classifications, the center of rotation possesses two degrees of freedom. Dihedral groups additionally feature an extra degree of freedom pertaining to the mirror positions.

The other two-dimensional isometry groups that possess a fixed point include:

• The special orthogonal group SO(2), which comprises all rotations around a fixed point. This group is also known as the circle group S¹, representing the multiplicative group of complex numbers with an absolute value of 1. It constitutes the proper symmetry group of a circle and serves as the continuous analogue of C_n. No geometric figure possesses the circle group as its full symmetry group; however, this may apply to a vector field.
• The orthogonal group O(2), which includes all rotations about a fixed point and reflections across any axis passing through that point. This group represents the symmetry of a circle. It is also referred to as Dih(S¹), being the generalized dihedral group of S§23§.

Isometry groups for non-bounded figures may incorporate translations, and these groups include:

• The seven frieze groups.
• The seventeen wallpaper groups.
• For each one-dimensional symmetry group, the combination of all symmetries within that group along one direction, coupled with the group of all translations in the perpendicular direction.
• The preceding description also applies when reflections in a line along the initial direction are included.

Three Dimensions

Up to conjugacy, the set of three-dimensional point groups comprises 7 infinite series and 7 distinct individual groups. In crystallography, only those point groups that preserve a crystal lattice are considered, meaning their rotations are restricted to orders 1, 2, 3, 4, or 6. This crystallographic constraint, applied to the infinite families of general point groups, yields 32 crystallographic point groups, consisting of 27 individual groups derived from the 7 series and 5 of the 7 other individual groups.

The continuous symmetry groups possessing a fixed point encompass those exhibiting:

• Cylindrical symmetry without a symmetry plane perpendicular to the axis, exemplified by objects such as a bottle or a cone.
• Cylindrical symmetry with a symmetry plane perpendicular to the axis.
• Spherical symmetry.

In the context of scalar field patterns, cylindrical symmetry inherently entails vertical reflection symmetry. Conversely, this implication does not hold for vector field patterns. For instance, considering a vector field expressed in cylindrical coordinates relative to a specific axis, the vector field ${\displaystyle \mathbf {A} =A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}}$ exhibits cylindrical symmetry about the axis if and only if ${\displaystyle A_{\rho },A_{\phi },}$ and ${\displaystyle A_{z}}$ demonstrate this symmetry (i.e., they are independent of ${\displaystyle \phi }$); furthermore, it possesses reflectional symmetry exclusively when A ϕ<!-- ϕ --> = §177178§ {\displaystyle A_{\phi }=0} .

For spherical symmetry, such a distinction does not apply, as all patterned objects inherently possess planes of reflection symmetry.

The continuous symmetry groups without a fixed point encompass groups characterized by a screw axis, for example, an infinite helix.

General Aspects of Symmetry Groups

Within broader mathematical frameworks, a symmetry group can be conceptualized as any form of transformation group or automorphism group. Every distinct mathematical structure is associated with invertible mappings that maintain its inherent properties. Conversely, the precise definition of a symmetry group can either delineate the structure itself or, at minimum, elucidate the concepts of geometric congruence or invariance. This perspective aligns with a fundamental tenet of the Erlangen Programme.

For instance, entities within a hyperbolic non-Euclidean geometry exhibit Fuchsian symmetry groups, which constitute the discrete subgroups of the hyperbolic plane's isometry group, maintaining hyperbolic distance instead of Euclidean distance. Certain examples are illustrated in Escher's artwork. Analogously, automorphism groups associated with finite geometries conserve collections of point-sets (discrete subspaces) rather than Euclidean subspaces, distances, or inner products. Consistent with Euclidean figures, objects situated in any geometric space possess symmetry groups that are subgroups of the ambient space's symmetries.

A further illustration of a symmetry group is found in combinatorial graphs, where a graph symmetry is defined as a permutation of vertices that maps edges to other edges. Every finitely presented group corresponds to the symmetry group of its Cayley graph, while the free group represents the symmetry group of an infinite tree graph.

Elucidating Group Structure via Symmetries

Cayley's theorem posits that every abstract group functions as a subgroup of the permutations of a specific set X, thereby allowing its conceptualization as the symmetry group of X augmented with additional structure. Furthermore, numerous abstract characteristics of a group, which are defined exclusively by its group operation, can be elucidated through the lens of symmetries.

As an illustration, consider G = Sym(X) as the finite symmetry group of a geometric figure X within a Euclidean space, and let H ⊂ G denote a subgroup. In this context, H can be understood as the symmetry group of X⁺, which represents a "decorated" variant of X. This decoration can be achieved by introducing patterns, such as arrows or colors, to X in a manner that eliminates all inherent symmetry, resulting in a figure X^# where Sym(X^#) = {1}, signifying the trivial subgroup. This implies that gX^# ≠ X^# for every non-trivial element g ∈ G. Consequently, the following relationship is established:

${\displaystyle X^{+}\ =\ \bigcup _{h\in H}hX^{\#}\quad {\text{satisfies}}\quad H=\mathrm {Sym} (X^{+}).}$

Within this conceptual framework, normal subgroups can also be precisely characterized. The symmetry group corresponding to the translation gX ⁺ is identified as the conjugate subgroup gHg⁻¹. Consequently, H is deemed normal under the following condition:

${\displaystyle \mathrm {Sym} (gX^{+})=\mathrm {Sym} (X^{+})\ \ {\text{for all}}\ g\in G;}$

This condition implies that the embellishment of X⁺ can be depicted in any orientation, relative to any aspect or characteristic of X, while consistently preserving the identical symmetry group gHg⁻¹ = H.

For instance, consider the dihedral group G = D§45§ = Sym(X), where X represents an equilateral triangle. This figure can be adorned with an arrow on one edge, resulting in an asymmetric configuration X^#. If τ ∈ G represents the reflection across the arrowed edge, the resultant composite figure X⁺ = X^# ∪ τX^# features a bidirectional arrow on that edge, and its corresponding symmetry group is H = {1, τ}. This subgroup lacks normality because gX⁺ could exhibit the bidirectional arrow on an alternative edge, thereby yielding a distinct reflection symmetry group.

Conversely, if H = {1, ρ, ρ²} ⊂ D§67§ denotes the cyclic subgroup generated by a rotational operation, the decorated figure X⁺ comprises a 3-cycle of arrows maintaining a consistent orientation. Consequently, H is normal, as depicting such a cycle with either rotational sense results in the identical symmetry group H.

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