This articledoes notcite anysources. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged andremoved. Find sources: "One-dimensional symmetry group" – news ·newspapers ·books ·scholar ·JSTOR(December 2009) (Learn how and when to remove this message)

Aone-dimensional symmetry group is amathematical group that describessymmetries in one dimension (1D).

A pattern in 1D can be represented as a functionf(x) for, say, the color at positionx.

The only nontrivial point group in 1D is a simplereflection. It can be represented by the simplestCoxeter group, A₁, [ ], orCoxeter-Dynkin diagram.

Affine symmetry groups representtranslation. Isometries which leave the function unchanged aretranslationsx +a witha such thatf(x +a) =f(x) andreflectionsa −x with a such thatf(a −x) =f(x). The reflections can be represented by theaffine Coxeter group [∞], orCoxeter-Dynkin diagram representing two reflections, and the translational symmetry as [∞]⁺, or Coxeter-Dynkin diagram as the composite of two reflections.

For a pattern without translational symmetry there are the following possibilities (1Dpoint groups):

• the symmetry group is the trivial group (no symmetry)
• the symmetry group is one of the groups each consisting of the identity and reflection in a point (isomorphic toZ₂)

Group	Coxeter		Description
C1	[ ]+		Identity,Trivial group Z1
D1	[ ]		Reflection.Abstract groups Z2 or Dih1.

These affine symmetries can be considered limiting cases of the2D dihedral and cyclic groups:

Group	Coxeter		Description
C∞	[∞]+		Cyclic: ∞-fold rotations become translations. Abstract group Z∞, theinfinite cyclic group.
D∞	[∞]		Dihedral: ∞-fold reflections. Abstract group Dih∞, theinfinite dihedral group.

Consider all patterns in 1D which have translationalsymmetry, i.e., functionsf(x) such that for somea > 0,f(x +a) =f(x) for allx. For these patterns, the values ofa for which this property holds form agroup.

We first consider patterns for which the group isdiscrete, i.e., for which the positive values in the group have a minimum. By rescaling we make this minimum value 1.

Such patterns fall in two categories, the two 1Dspace groups orline groups.

In the simpler case the only isometries ofR which map the pattern to itself are translations; this applies, e.g., for the pattern

− −−−  − −−−  − −−−  − −−−

Each isometry can be characterized by an integer, namely plus or minus the translation distance. Therefore thesymmetry group isZ.

In the other case, among the isometries ofR which map the pattern to itself there are also reflections; this applies, e.g., for the pattern

− −−− −  − −−− −  − −−− −

We choose the origin forx at one of the points of reflection. Now all reflections which map the pattern to itself are of the forma−x where the constant "a" is an integer (the increments ofa are 1 again, because we can combine a reflection and a translation to get another reflection, and we can combine two reflections to get a translation). Therefore all isometries can be characterized by an integer and a code, say 0 or 1, for translation or reflection.

Thus:

• ${\displaystyle (a,0):x\mapsto x+a}$
• ${\displaystyle (a,1):x\mapsto a-x}$

The latter is a reflection with respect to the pointa/2 (an integer or an integer plus 1/2).

Group operations (function composition, the one on the right first) are, for integersa andb:

• ${\displaystyle (a,0)\circ (b,0)=(a+b,0)}$
• ${\displaystyle (a,0)\circ (b,1)=(a+b,1)}$
• ${\displaystyle (a,1)\circ (b,0)=(a-b,1)}$
• ${\displaystyle (a,1)\circ (b,1)=(a-b,0)}$

E.g., in the third case: translation by an amountb changesx intox +b, reflection with respect to 0 gives−x −b, and a translationa givesa −b −x.

This group is called thegeneralized dihedral group ofZ, Dih(Z), and also D_∞. It is asemidirect product ofZ and C₂. It has anormal subgroup ofindex 2 isomorphic toZ: the translations. Also it contains an elementf of order 2 such that, for alln inZ,  n f = f n ⁻¹: the reflection with respect to the reference point, (0,1).

The two groups are calledlattice groups. Thelattice isZ. As translation cell we can take the interval 0 ≤x < 1. In the first case thefundamental domain can be taken the same; topologically it is a circle (1-torus); in the second case we can take 0 ≤x ≤ 0.5.

The actualdiscrete symmetry group of a translationally symmetric pattern can be:

• of group 1 type, for any positive value of the smallest translation distance
• of group 2 type, for any positive value of the smallest translation distance, and any positioning of the lattice of points of reflection (which is twice as dense as the translation lattice)

The set of translationally symmetric patterns can thus be classified by actual symmetry group, while actual symmetry groups, in turn, can be classified as type 1 or type 2.

These space group types are the symmetry groups “up to conjugacy with respect to affine transformations”: the affine transformation changes the translation distance to the standard one (above: 1), and the position of one of the points of reflections, if applicable, to the origin. Thus the actual symmetry group contains elements of the formgag⁻¹=b, which is a conjugate ofa.

For a homogeneous “pattern” the symmetry group contains all translations, and reflection in all points. The symmetry group is isomorphic to Dih(R).

There are also less trivial patterns/functions with translational symmetry for arbitrarily small translations, e.g. the group of translations by rational distances. Even apart from scaling and shifting, there are infinitely many cases, e.g. by considering rational numbers of which the denominators are powers of a given prime number.

The translations form a group of isometries. However, there is no pattern with this group as symmetry group.

Symmetries of a function (in the sense of this article) imply corresponding symmetries of its graph. However, 2-fold rotational symmetry of the graph does not imply any symmetry (in the sense of this article) of the function: function values (in a pattern representing colors, grey shades, etc.) arenominal data, i.e. grey is not between black and white, the three colors are simply all different.

Even with nominal colors there can be a special kind of symmetry, as in:

−−−−−−− -- − −−−   − −  −

(reflection gives the negative image). This is also not included in the classification.

Group actions of the symmetry group that can be considered in this connection are:

• onR
• on the set of real functions of a real variable (each representing a pattern)

This section illustrates group action concepts for these cases.

The action ofG onX is called

• transitive if for any twox,y inX there exists ag inG such thatg ·x =y; for neither of the two group actions this is the case for any discrete symmetry group
• faithful (oreffective) if for any two differentg,h inG there exists anx inX such thatg ·x ≠h ·x; for both group actions this is the case for any discrete symmetry group (because, except for the identity, symmetry groups do not contain elements that “do nothing”)
• free if for any two differentg,h inG and allx inX we haveg ·x ≠h ·x; this is the case if there are no reflections
• regular (orsimply transitive) if it is both transitive and free; this is equivalent to saying that for any twox,y inX there exists precisely oneg inG such thatg ·x =y.

Consider a groupG acting on a setX. Theorbit of a pointx inX is the set of elements ofX to whichx can be moved by the elements ofG. The orbit ofx is denoted byGx:

${\displaystyle Gx=\left\{g\cdot x\mid g\in G\right\}.}$

Case that the group action is onR:

• For the trivial group, all orbits contain only one element; for a group of translations, an orbit is e.g. {..,−9,1,11,21,..}, for a reflection e.g. {2,4}, and for the symmetry group with translations and reflections, e.g., {−8,−6,2,4,12,14,22,24,..} (translation distance is 10, points of reflection are ..,−7,−2,3,8,13,18,23,..). The points within an orbit are “equivalent”. If a symmetry group applies for a pattern, then within each orbit the color is the same.

Case that the group action is on patterns:

• The orbits are sets of patterns, containing translated and/or reflected versions, “equivalent patterns”. A translation of a pattern is only equivalent if the translation distance is one of those included in the symmetry group considered, and similarly for a mirror image.

The set of all orbits ofX under the action ofG is written asX/G.

IfY is asubset ofX, we writeGY for the set {g ·y :y${\displaystyle \in }$Y andg${\displaystyle \in }$G}. We call the subsetYinvariant under G ifGY =Y (which is equivalent toGY ⊆Y). In that case,G also operates onY. The subsetY is calledfixed under G ifg ·y =yfor allg inG and ally inY. In the example of the orbit {−8,−6,2,4,12,14,22,24,..}, {−9,−8,−6,−5,1,2,4,5,11,12,14,15,21,22,24,25,..} is invariant underG, but not fixed.

For everyx inX, we define thestabilizer subgroup ofx (also called theisotropy group orlittle group) as the set of all elements inG that fixx:

${\displaystyle G_x=\{g\in G\mid g\cdot x=x\}.}$

Ifx is a reflection point, its stabilizer is the group of order two containing the identity and the reflection inx. In other cases the stabilizer is the trivial group.

For a fixedx inX, consider the map fromG toX given by${\displaystyle g\mid \to g\cdot x}$. Theimage of this map is the orbit ofx and thecoimage is the set of all leftcosets ofG_x. The standard quotient theorem of set theory then gives a naturalbijection between${\displaystyle G/G_x}$ and${\displaystyle Gx}$. Specifically, the bijection is given by${\displaystyle h{G}_x\mid \to h\cdot x}$. This result is known as theorbit-stabilizer theorem. If, in the example, we take${\displaystyle x=3}$, the orbit is {−7,3,13,23,..}, and the two groups are isomorphic withZ.

If two elements${\displaystyle x}$ and${\displaystyle y}$ belong to the same orbit, then their stabilizer subgroups,${\displaystyle G_x}$ and${\displaystyle G_y}$, areisomorphic. More precisely: if${\displaystyle y=g\cdot x}$, then${\displaystyle G_y=g{G}_x{g}^{-1}}$. In the example this applies e.g. for 3 and 23, both reflection points. Reflection about 23 corresponds to a translation of −20, reflection about 3, and translation of 20.

• Line group
• Frieze group
• Space group
• Wallpaper group

• Euclidean geometry
• Group theory
• Symmetry
• 1 (number)

• Articles with short description
• Short description matches Wikidata
• Articles lacking sources from December 2009
• All articles lacking sources