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

- Permutations of a set of three elements
- The group of translations of the plane
- The symmetry group of a square: dihedral group of order 8
- Generating the group
- Normal subgroup
- Free group on two generators
- The set of maps
- The sets of maps from a set to a group
- Automorphism groups
- Groups of permutations
- Matrix groups
- See also
- References

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:

*e*: RGB → RGB*a*: RGB → GRB*b*: RGB → RBG*ab*: RGB → BRG*ba*: RGB → GBR*aba*: RGB → BGR

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 * S_{3}, has order 6, and is non-abelian (since, for example, *ab* ≠ *ba*).

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 *a* ∘ *b* 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

*a*∘*b*= "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

*a*∘*b*= "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:

- If
*a*and*b*are translations, then*a*∘*b*is also a translation. - Composition of translations is associative: (
*a*∘*b*) ∘*c*=*a*∘ (*b*∘*c*). - The identity element for this group is the translation with prescription "move zero miles in any direction".
- 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.

Dih _{4} as 2D point group, D_{4}, [4], (*4•), order 4, with a 4-fold rotation and a mirror generator. | Dih _{4} in 3D dihedral group D_{4}, [4,2]^{+}, (422), order 4, with a vertical 4-fold rotation generator order 4, and 2-fold horizontal generator |

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 *x* ∘ *y* 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.

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 *a* ∘ *a*, also written as *a*^{2}, is a 180° degree turn. *a*^{3} is a 270° clockwise rotation (or a 90° counter-clockwise rotation). We also see that *b*^{2} = *e* and also *a*^{4} = *e*. Here's an interesting one: what does *a* ∘ *b* do? First flip horizontally, then rotate. Try to visualize that *a* ∘ *b* = *b* ∘ *a*^{3}. Also, *a*^{2} ∘ *b* is a vertical flip and is equal to *b* ∘ *a*^{2}.

We say that elements *a* and *b* generate the group.

This group of order 8 has the following Cayley table:

∘ | e | b | a | a^{2} | a^{3} | ab | a^{2}b | a^{3}b |
---|---|---|---|---|---|---|---|---|

e | e | b | a | a^{2} | a^{3} | ab | a^{2}b | a^{3}b |

b | b | e | a^{3}b | a^{2}b | ab | a^{3} | a^{2} | a |

a | a | ab | a^{2} | a^{3} | e | a^{2}b | a^{3}b | b |

a^{2} | a^{2} | a^{2}b | a^{3} | e | a | a^{3}b | b | ab |

a^{3} | a^{3} | a^{3}b | e | a | a^{2} | b | ab | a^{2}b |

ab | ab | a | b | a^{3}b | a^{2}b | e | a^{3} | a^{2} |

a^{2}b | a^{2}b | a^{2} | ab | b | a^{3}b | a | e | a^{3} |

a^{3}b | a^{3}b | a^{3} | a^{2}b | ab | b | a^{2} | a | e |

For any two elements in the group, the table records what their composition is. Here we wrote "*a*^{3}*b*" as a shorthand for *a*^{3} ∘ *b*.

In mathematics this group is known as the ** dihedral group ** of order 8, and is either denoted Dih_{4}, D_{4} or D_{8}, 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.

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 D _{4}e r _{1}r _{2}r _{3}f _{v}f _{h}f _{d}f _{c}e e r _{1}r _{2}r _{3}f _{v}f _{h}f _{d}f _{c}r _{1}r _{1}r _{2}r _{3}e f _{c}f _{d}f _{v}f _{h}r _{2}r _{2}r _{3}e r _{1}f _{h}f _{v}f _{c}f _{d}r _{3}r _{3}e r _{1}r _{2}f _{d}f _{c}f _{h}f _{v}f _{v}f _{v}f _{d}f _{h}f _{c}e r _{2}r _{1}r _{3}f _{h}f _{h}f _{c}f _{v}f _{d}r _{2}e r _{3}r _{1}f _{d}f _{d}f _{h}f _{c}f _{v}r _{3}r _{1}e r _{2}f _{c}f _{c}f _{v}f _{d}f _{h}r _{1}r _{3}r _{2}e The elements e, r _{1}, r_{2}, and r_{3}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.

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*^{−1}*a*^{−1}" concatenated with "*abab*^{−1}*a*" yields "*abab*^{−1}*a*^{−1}*abab*^{−1}*a*", which gets reduced to "*abaab*^{−1}*a*". 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.

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*).

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 , Σ_{S}, 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.

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 GL_{n}(**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 **, SL_{n}(**R**) or SL(*n*, **R**). Geometrically, this consists of all the elements of GL_{n}(**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 ** O_{n}(**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 ** SO_{n}(**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] }

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.

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

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.

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.

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 Z_{2} × Z_{2}, 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

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:

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.

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.

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.

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.

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

**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.

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**

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.

In mathematics, **D _{3}** (sometimes alternatively denoted by

**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.

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.

**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.

- ↑ 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 .

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