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单词 ExamplesOfGroups
释义

examples of groups


Groups (http://planetmath.org/Group) are ubiquitous throughout mathematics. Many “naturallyoccurring” groups are either groups of numbers (typically AbelianMathworldPlanetmath)or groups of symmetriesMathworldPlanetmathPlanetmathPlanetmath (typically non-AbelianMathworldPlanetmathPlanetmath).

Groups of numbers

  • The most important group is the group of integers withadditionPlanetmathPlanetmath as operationMathworldPlanetmath and zero as identity elementMathworldPlanetmath.

  • The integers modulo n, often denoted by n, form agroup under addition. Like itself, this is a cyclic groupMathworldPlanetmath; anycyclic group is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to one of these.

  • The rational (or real, or complex) numbers form a group underaddition.

  • The positive rationals form a group under multiplication with identity element 1, and sodo the non-zero rationals. The same is true for the reals and real algebraic numbersMathworldPlanetmath.

  • The non-zero complex numbersMathworldPlanetmathPlanetmath form a group under multiplication.So do the non-zero quaternionsMathworldPlanetmath. The latter is our first example of anon-Abelian groupMathworldPlanetmath.

  • More generally, any (skew) field gives rise to two groups: the additivegroupMathworldPlanetmath of all field elements with 0 as identity element, and the multiplicative group of allnon-zero field elements with 1 as identity element.

  • The complex numbers of absolute valueMathworldPlanetmathPlanetmathPlanetmath 1 form a group undermultiplication, best thought of as the unit circleMathworldPlanetmath. The quaternions ofabsolute value 1 form a group under multiplication, best thought of asthe three-dimensional unit sphereMathworldPlanetmath S3. The two-dimensional sphereS2 howeveris not a group in any natural way.

  • The positive integers less than n which arecoprimeMathworldPlanetmathPlanetmath to n form a group if the operation is defined asmultiplication modulo n. This is an Abelian group whose order is given by theEuler phi-function ϕ(n).

  • The units of the number ring [3] form the multiplicative group consisting of all integer powers of 2+3 and their negatives (see units of quadratic fields).

  • Generalizing the last two examples, if R is a ring with multiplicative identityPlanetmathPlanetmath 1, then the units of R (http://planetmath.org/GroupOfUnits) (the elements invertiblePlanetmathPlanetmathPlanetmathPlanetmath with respect to multiplication) form a group with respect to ring multiplication and with identity element 1. See examples of rings.

Most groups of numbers carry natural topologies turning them intotopological groups.

Symmetry groups

  • The symmetric groupMathworldPlanetmathPlanetmath of degree n, denoted by Sn, consists ofall permutationsMathworldPlanetmath of n items and has n! elements. Every finitegroupMathworldPlanetmath is isomorphic to a subgroupMathworldPlanetmathPlanetmath of some Sn (Cayley’s theorem).

  • An important subgroup of the symmetric group of degree n is thealternating groupMathworldPlanetmath, denoted An. This consists of all evenpermutations on n items. A permutation is said to be even if it canbe written as the productMathworldPlanetmathPlanetmath of an even numberMathworldPlanetmath of transpositionsMathworldPlanetmath. The alternatinggroup is normal in Sn, of index 2, and it is an interesting fact that Anis simple for n5. See the proof on the simplicity of the alternatinggroups. See also examples of finite simple groups.

  • If any geometrical object is given, one can consider itssymmetry group consisting of all rotationsMathworldPlanetmath and reflectionsMathworldPlanetmath which leavethe object unchanged. For example, the symmetry group of a cone isisomorphic to S1; the symmetry group of a square has eight elements and is isomorphic to the dihedral groupMathworldPlanetmath D4.

  • The set of all automorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of a given group (or field,or graph, or topological space, or object in any categoryMathworldPlanetmath) forms a group with operationgiven by the compositionMathworldPlanetmathPlanetmath of homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. This is the automorphism groupMathworldPlanetmath of the given object and captures its “internal symmetries”.

  • In Galois theoryMathworldPlanetmath, the symmetry groups of field extensions (orequivalently: the symmetry groups of solutions to polynomialequations) are the central object of study; they are called Galois groupsMathworldPlanetmath. One version of the inverse Galois problem asks whether every finite group can arise as the symmetry group of some algebraic extensionMathworldPlanetmath of the rational numbers. The answer is unknown.

  • Several matrix groupsMathworldPlanetmath describe various aspects of the symmetry of n-space:

    • The general linear groupMathworldPlanetmath GL(n,) of all real invertiblen×n matrices (with matrix multiplicationMathworldPlanetmath as operation) containsrotations, reflections, dilationsMathworldPlanetmath, shear transformations, and theircombinationsMathworldPlanetmathPlanetmath.

    • The orthogonal groupMathworldPlanetmath O(n,) of all real orthogonalMathworldPlanetmathPlanetmathPlanetmathPlanetmathn×n matrices contains the rotations and reflections of n-space.

    • The special orthogonal groupMathworldPlanetmath SO(n,) of all real orthogonaln×nmatrices with determinantMathworldPlanetmath 1 contains the rotations of n-space.

    All these matrix groups are Lie groups: groups which are differentiable manifoldssuch that the group operationsMathworldPlanetmath are smooth maps.

Other groups

  • The trivial group consists only of its identity element.

  • The Klein 4-group is a non-cyclic abelian group with four elements. For other small groups, see groups of small order.

  • If X is a topological space and x is a point of X, we candefine the fundamental groupMathworldPlanetmath of X at x. It consists of(homotopy classes of) continuousMathworldPlanetmathPlanetmathpaths starting and ending at x and describes the structureMathworldPlanetmathof the “holes” in X accessiblePlanetmathPlanetmath from x. The fundamental group is generalized by the higher homotopy groups.

  • Other groups studied in algebraic topology are the homology groupsMathworldPlanetmath of a topological space. In a different way, they also provide information about the “holes” of the space.

  • The free groupsMathworldPlanetmath are important in algebraic topology. In a sense,they are the most general groups, having only those relationsMathworldPlanetmathPlanetmathPlanetmath amongtheir elements that are absolutely required by the group axioms. The free group on the set S has as members all the finite strings that can be formed from elements of S and their inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath; the operation comes from string concatenation.

  • If A and B are two Abelian groups (or modules over the same ring), then the set Hom(A,B) of all homomorphisms from A to B is an Abelian group. Note that the commutativity of B is crucial here: without it, one couldn’t prove that the sum of two homomorphisms is again a homomorphism.

  • Given any set X, the powerset 𝒫(X) of X becomes an abelian group if we use the symmetric differenceMathworldPlanetmathPlanetmath as operation. In this group, any element is its own inverse, which makes it into a vector spaceMathworldPlanetmath over 2.

  • If R is a ring with multiplicative identity, then the set of all invertible n×n matrices over Rforms a group under matrix multiplication with the identity matrixMathworldPlanetmath as identity element; this group is denoted by GL(n,R). It is the group of units of the ring of all n×n matrices over R. For a given n, the groups GL(n,R) with commutative ring R can be viewed as the points on the general linear group scheme GLn.

  • If K is a number fieldMathworldPlanetmath, then multiplication of (equivalence classesMathworldPlanetmathPlanetmath of) non-zero ideals in the ring of algebraic integers 𝒪𝒦 gives rise to the ideal class groupPlanetmathPlanetmathPlanetmath of K.

  • The set of the equivalence classes of commensurability of the positive real numbers is an Abelian group with respect to the defined operation.

  • The set of arithmetic functions that take a value other than 0 at 1 form an Abelian group under Dirichlet convolution. They include as a subgroup the set of multiplicative functions.

  • Consider the curve C={(x,y)K2y2=x3-x}, where K is any field. Every straight line intersects this set in three points (counting a point twice if the line is tangentMathworldPlanetmathPlanetmathPlanetmath, and allowing for a point at infinity). If we require that those three points add up to zero for any straight line, then we have defined an abelian group structure on C. Groups like these are called abelian varietiesMathworldPlanetmath.

  • Let E be an elliptic curveMathworldPlanetmath defined over any field F. Then the set of F-rational points in the curve E, denoted by E(F), can be given the structure of abelian group. If F is a number field, then E(F) is a finitely generatedMathworldPlanetmathPlanetmathPlanetmath abelian group. The curve C in the example above is an elliptic curve defined over , thus C() is a finitely generated abelian group.

  • In the classification of all finite simple groups, several“sporadic” groups occur which don’t follow any discernable pattern.The largest of these is the monster group with about 81053elements.

Titleexamples of groups
Canonical nameExamplesOfGroups
Date of creation2013-03-22 12:49:19
Last modified on2013-03-22 12:49:19
OwnerAxelBoldt (56)
Last modified byAxelBoldt (56)
Numerical id34
AuthorAxelBoldt (56)
Entry typeExample
Classificationmsc 20-00
Classificationmsc 20A05
Related topicExamplesOfFiniteSimpleGroups
Related topicSpinGroup
Related topicExamplesOfAlgebraicKTheoryGroups
Related topicQuantumGroups
Related topicGroupsOfSmallOrder
Related topicTriangleGroups
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