Some elementary examples of groups in
mathematics are given on
Group (mathematics)
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. ...
.
Further examples are listed here.
Permutations of a set of three elements
![Group diagram d6](https://upload.wikimedia.org/wikipedia/commons/0/0a/Group_diagram_d6.svg)
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
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
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
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
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 ''
generates'' this group. The group, called the ''
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
'' S
3, has
order 6, and is
non-abelian (since, for example, ''ab'' ≠ ''ba'').
The group of translations of the plane
A ''translation'' of the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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
![Dih 4 Cayley Graph; generators b, c](https://upload.wikimedia.org/wikipedia/commons/f/f5/Dih_4_Cayley_Graph%3B_generators_b%2C_c.svg)
Groups are very important to describe the
symmetry of objects, be they
geometrical
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
(like a
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
) or
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
ic (like a set of equations).
As an example, we consider a glass
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
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
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
s.
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 ''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 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 multiplic ...
:
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
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, ...
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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, which can be seen from the table; the table is not symmetrical about the main diagonal.
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
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
is normal, the left coset is the same as the right coset.
:
Free group on two generators
The
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
with two generators ''a'' and ''b'' consists of all finite
strings
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian anim ...
/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
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
; the free group in two generators is also used for a
proof of the
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...
.
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
mappings of a set ''S'' onto itself. Then ''G'' forms a group under ordinary
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of mappings. This group is called the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
, and is commonly denoted
, Σ
''S'', or
. The identity element of ''G'' is the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
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
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
.
Matrix groups
If ''n'' is some positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, we can consider the set of all
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
''n'' by ''n''
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
components, say.
This is a group with
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
as the operation. It is called the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
that
fix a given point (the origin).
If we restrict ourselves to matrices with
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1, then we get another group, the
special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
, 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 solid
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
s in Euclidean space.
If instead we restrict ourselves to
orthogonal matrices
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 identity ma ...
, 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 group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fa ...
s.
If this idea is generalised to matrices with
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s as entries, then we get further useful Lie groups, such as the
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
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
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
Sp(''n'').
Furthermore, the idea can be treated purely algebraically with matrices over any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, but then the groups are not Lie groups.
For example, we have the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
s over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s. The group theorist
J. L. Alperin
Jonathan Lazare Alperin (; born 1937) is an Americans, American mathematician specializing in the area of algebra known as group theory. He is notable for his work in group theory which has been cited over 500 times according to the Mathematical R ...
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."
See also
*
List of small groups
The following list in mathematics contains the finite groups of small order up to group isomorphism.
Counts
For ''n'' = 1, 2, … the number of nonisomorphic groups of order ''n'' is
: 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5 ...
*
List of the 230 crystallographic 3D space groups
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchan ...
References
{{reflist
Group theory
Groups
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 ide ...