Examples of groups

Last updated

Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here.

Contents

Permutations of a set of three elements

Cycle graph for S3. A loop specifies a series of powers of any element connected to the identity element (e). For example, the e-ba-ab loop reflects the fact that ba = ab and ba = e, as well as the fact that ab = ba and ab = e. The other "loops" are roots of unity so that, for example a = e. Group diagram d6.svg
Cycle graph for S3. A loop specifies a series of powers of any element connected to the identity element (e). For example, the e-ba-ab loop reflects the fact that ba = ab and ba = e, as well as the fact that ab = ba and ab = e. The other "loops" are roots of unity so that, for example a = e.

Consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the operation "swap the first block and the second block", and b be the operation "swap the second block and the third block".

We can write xy for the operation "first do y, then do x"; so that ab is the operation RGB → RBG → BRG, which could be described as "move the first two blocks one position to the right and put the third block into the first position". If we write e for "leave the blocks as they are" (the identity operation), then we can write the six permutations of the three blocks as follows:

Note that aa has the effect RGB → GRB → RGB; so we can write aa = e. Similarly, bb = (aba)(aba) = e; (ab)(ba) = (ba)(ab) = e; so every element has an inverse.

By inspection, we can determine associativity and closure; note in particular that (ba)b = bab = b(ab).

Since it is built up from the basic operations a and b, we say that the set {a,b} generates this group. The group, called the symmetric group S3, has order 6, and is non-abelian (since, for example, abba).

The group of translations of the plane

A translation of the plane is a rigid movement of every point of the plane for a certain distance in a certain direction. For instance "move in the North-East direction for 2 miles" is a translation of the plane. Two translations such as a and b can be composed to form a new translation ab as follows: first follow the prescription of b, then that of a. For instance, if

a = "move North-East for 3 miles"

and

b = "move South-East for 4 miles"

then

ab = "move to bearing 8.13° for 5 miles" (bearing is measured counterclockwise and from East)

Or, if

a = "move to bearing 36.87° for 3 miles" (bearing is measured counterclockwise and from East)

and

b = "move to bearing 306.87° for 4 miles" (bearing is measured counterclockwise and from East)

then

ab = "move East for 5 miles"

(see Pythagorean theorem for why this is so, geometrically).

The set of all translations of the plane with composition as the operation forms a group:

  1. If a and b are translations, then ab is also a translation.
  2. Composition of translations is associative: (ab)c = a(bc).
  3. The identity element for this group is the translation with prescription "move zero miles in any direction".
  4. The inverse of a translation is given by walking in the opposite direction for the same distance.

This is an abelian group and our first (nondiscrete) example of a Lie group: a group which is also a manifold.

The symmetry group of a square: dihedral group of order 8

Cycle graph of Dih4
a is the clockwise rotation
and b the horizontal reflection. Dih4 cycle graph.svg
Cycle graph of Dih4
a is the clockwise rotation
and b the horizontal reflection.
Dihedral group4 example.png
Dih4 as 2D point group, D4, [4], (*4•), order 4, with a 4-fold rotation and a mirror generator.
Dihedral group4 example2.png
Dih4 in 3D dihedral group D4, [4,2]+, (422), order 4, with a vertical 4-fold rotation generator order 4, and 2-fold horizontal generator
Cayley graph of Dih4 Dih 4 Cayley Graph; generators a, b.svg
Cayley graph of Dih4
A different Cayley graph of Dih4, generated by the horizontal reflection b and a diagonal reflection c Dih 4 Cayley Graph; generators b, c.svg
A different Cayley graph of Dih4, generated by the horizontal reflection b and a diagonal reflection c

Groups are very important to describe the symmetry of objects, be they geometrical (like a tetrahedron) or algebraic (like a set of equations). As an example, we consider a glass square of a certain thickness (with a letter "F" written on it, just to make the different positions distinguishable).

In order to describe its symmetry, we form the set of all those rigid movements of the square that don't make a visible difference (except the "F"). For instance, if an object turned 90° clockwise still looks the same, the movement is one element of the set, for instance a. We could also flip it around a vertical axis so that its bottom surface becomes its top surface, while the left edge becomes the right edge. Again, after performing this movement, the glass square looks the same, so this is also an element of our set and we call it b. The movement that does nothing is denoted by e.

Given two such movements x and y, it is possible to define the composition xy as above: first the movement y is performed, followed by the movement x. The result will leave the slab looking like before.

The point is that the set of all those movements, with composition as the operation, forms a group. This group is the most concise description of the square's symmetry. Chemists use symmetry groups of this type to describe the symmetry of crystals and molecules.

Generating the group

Let's investigate our square's symmetry group some more. Right now, we have the elements a, b and e, but we can easily form more: for instance aa, also written as a2, is a 180° degree turn. a3 is a 270° clockwise rotation (or a 90° counter-clockwise rotation). We also see that b2 = e and also a4 = e. Here's an interesting one: what does ab do? First flip horizontally, then rotate. Try to visualize that ab = ba3. Also, a2b is a vertical flip and is equal to ba2.

We say that elements a and b generate the group.

This group of order 8 has the following Cayley table:

ebaa2a3aba2ba3b
eebaa2a3aba2ba3b
bbea3ba2baba3a2a
aaaba2a3ea2ba3bb
a2a2a2ba3eaa3bbab
a3a3a3beaa2baba2b
abababa3ba2bea3a2
a2ba2ba2abba3baea3
a3ba3ba3a2babba2ae

For any two elements in the group, the table records what their composition is. Here we wrote "a3b" as a shorthand for a3b.

In mathematics this group is known as the dihedral group of order 8, and is either denoted Dih4, D4 or D8, depending on the convention. This was an example of a non-abelian group: the operation ∘ here is not commutative, which can be seen from the table; the table is not symmetrical about the main diagonal.

The dihedral group of order 8 is isomorphic to the permutation group generated by (1234) and (13). The numbers in this table come from numbering the 4! = 24 permutations of S4, which Dih4 is a subgroup of, from 0 (shown as a black circle) to 23. Dihedral group of order 8; Cayley table (element orders 1,2,2,4,2,2,4,2); subgroup of S4.svg
The dihedral group of order 8 is isomorphic to the permutation group generated by (1234) and (13). The numbers in this table come from numbering the 4! = 24 permutations of S4, which Dih4 is a subgroup of, from 0 (shown as a black circle) to 23.

Normal subgroup

This version of the Cayley table shows that this group has one normal subgroup shown with a red background. In this table r means rotations, and f means flips. Because the subgroup is normal, the left coset is the same as the right coset.

Group table of D4
er1r2r3fvfhfdfc
eer1r2r3fvfhfdfc
r1r1r2r3efcfdfvfh
r2r2r3er1fhfvfcfd
r3r3er1r2fdfcfhfv
fvfvfdfhfcer2r1r3
fhfhfcfvfdr2er3r1
fdfdfhfcfvr3r1er2
fcfcfvfdfhr1r3r2e
The elements e, r1, r2, and r3 form a subgroup, highlighted in   red (upper left region). A left and right coset of this subgroup is highlighted in   green (in the last row) and   yellow (last column), respectively.

Free group on two generators

The free group with two generators a and b consists of all finite strings/words that can be formed from the four symbols a, a−1, b and b−1 such that no a appears directly next to an a−1 and no b appears directly next to a b−1. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance: "abab−1a−1" concatenated with "abab−1a" yields "abab−1a−1abab−1a", which gets reduced to "abaab−1a". One can check that the set of those strings with this operation forms a group with the empty string ε := "" being the identity element (Usually the quotation marks are left off; this is why the symbol ε is required).

This is another infinite non-abelian group.

Free groups are important in algebraic topology; the free group in two generators is also used for a proof of the Banach–Tarski paradox.

The set of maps

The sets of maps from a set to a group

Let G be a group and S a set. The set of maps M(S, G) is itself a group; namely for two maps f,g of S into G we define fg to be the map such that (fg)(x) = f(x)g(x) for every x in S and f−1 to be the map such that f−1(x) = f(x)−1.

Take maps f, g, and h in M(S, G). For every x in S, f(x) and g(x) are both in G, and so is (fg)(x). Therefore, fg is also in M(S, G), i.e. M(S, G) is closed. M(S, G) is associative because ((fg)h)(x) = (fg)(x)h(x) = (f(x)g(x))h(x) = f(x)(g(x)h(x)) = f(x)(gh)(x) = (f(gh))(x). And there is a map i such that i(x) = e where e is the identity element of G. The map i is such that for all f in M(S, G) we have fi = if = f, i.e. i is the identity element of M(S, G). Thus, M(S, G) is actually a group.

If G is abelian then (fg)(x) = f(x)g(x) = g(x)f(x) = (gf)(x), and therefore so is M(S, G).

Automorphism groups

Groups of permutations

Let G be the set of bijective mappings of a set S onto itself. Then G forms a group under ordinary composition of mappings. This group is called the symmetric group , and is commonly denoted , ΣS, or . The identity element of G is the identity map of S. For two maps f,g in G are bijective, fg is also bijective. Therefore, G is closed. The composition of maps is associative; hence G is a group. S may be either finite or infinite.

Matrix groups

If n is some positive integer, we can consider the set of all invertible n by n matrices with real number components, say. This is a group with matrix multiplication as the operation. It is called the general linear group , and denoted GLn(R) or GL(n, R) (where R is the set of real numbers). Geometrically, it contains all combinations of rotations, reflections, dilations and skew transformations of n-dimensional Euclidean space that fix a given point (the origin).

If we restrict ourselves to matrices with determinant 1, then we get another group, the special linear group , SLn(R) or SL(n, R). Geometrically, this consists of all the elements of GLn(R) that preserve both orientation and volume of the various geometric solids in Euclidean space.

If instead we restrict ourselves to orthogonal matrices, then we get the orthogonal group On(R) or O(n, R). Geometrically, this consists of all combinations of rotations and reflections that fix the origin. These are precisely the transformations which preserve lengths and angles.

Finally, if we impose both restrictions, then we get the special orthogonal group SOn(R) or SO(n, R), which consists of rotations only.

These groups are our first examples of infinite non-abelian groups. They are also happen to be Lie groups. In fact, most of the important Lie groups (but not all) can be expressed as matrix groups.

If this idea is generalised to matrices with complex numbers as entries, then we get further useful Lie groups, such as the unitary group U(n). We can also consider matrices with quaternions as entries; in this case, there is no well-defined notion of a determinant (and thus no good way to define a quaternionic "volume"), but we can still define a group analogous to the orthogonal group, the symplectic group Sp(n).

Furthermore, the idea can be treated purely algebraically with matrices over any field, but then the groups are not Lie groups.

For example, we have the general linear groups over finite fields. The group theorist J. L. Alperin has written that "The typical example of a finite group is GL(n, q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled." [1]

See also

Related Research Articles

<span class="mw-page-title-main">Lie group</span> Group that is also a differentiable manifold with group operations that are smooth

In mathematics, a Lie group is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be a group, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.

<span class="mw-page-title-main">Linear algebra</span> Branch of mathematics

Linear algebra is the branch of mathematics concerning linear equations such as:

<span class="mw-page-title-main">Group (mathematics)</span> Set with associative invertible operation

In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.

<span class="mw-page-title-main">Group theory</span> Branch of mathematics that studies the properties of groups

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

<span class="mw-page-title-main">Klein four-group</span>

In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise exclusive or operations on two-bit binary values, or more abstractly as Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V or as K4.

<span class="mw-page-title-main">Semidirect product</span> Operation in group theory

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:

<span class="mw-page-title-main">Ring (mathematics)</span> Algebraic structure with addition and multiplication

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

<span class="mw-page-title-main">Dihedral group</span> Group of symmetries of a regular polygon

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes 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.

<span class="mw-page-title-main">Orthogonal group</span> Type of group in mathematics

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.

<span class="mw-page-title-main">Translation (geometry)</span> Planar movement within a Euclidean space without rotation

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free-modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.

<span class="mw-page-title-main">Glossary of group theory</span>

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.

<span class="mw-page-title-main">Rotation (mathematics)</span> Motion of a certain space that preserves at least one point

Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have sign : a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space.

<span class="mw-page-title-main">Circle group</span> Lie group of complex numbers of unit modulus; topologically a circle

In mathematics, the circle group, denoted by or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers

<span class="mw-page-title-main">Abstract polytope</span> Poset representing certain properties of a polytope

In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.

Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group – such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center – can be discovered from its Cayley table.

<span class="mw-page-title-main">Dihedral group of order 6</span> Non-commutative group with 6 elements

In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of order 6. It is isomorphic to the symmetric group S3 of degree 3. It is also the smallest possible non-abelian group.

<span class="mw-page-title-main">Symmetry in mathematics</span> Symmetry in mathematics

Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.

<span class="mw-page-title-main">Point groups in two dimensions</span>

In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself. That group is isomorphic to R/Z and the first unitary group, U(1), a group also known as the circle group.

<span class="mw-page-title-main">Molecular symmetry</span> Symmetry of chemical molecules

Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain many of a molecule's chemical properties, such as whether or not it has a dipole moment, as well as its allowed spectroscopic transitions. To do this it is necessary to use group theory. This involves classifying the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Symmetry is useful in the study of molecular orbitals, with applications to the Hückel method, to ligand field theory, and to the Woodward-Hoffmann rules. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry discuss symmetry. Another framework on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials.

References

  1. Alperin, Jonathan L. (1984). "Book Review: Finite groups". Bulletin of the American Mathematical Society. New Series. 10: 121–124. doi: 10.1090/S0273-0979-1984-15210-8 .