examples of groups

Groups (http://planetmath.org/Group) are ubiquitous throughout mathematics. Many “naturally occurring” 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 with additionPlanetmathPlanetmath as operationMathworldPlanetmath and zero as identity elementMathworldPlanetmath.

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

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

  • The positive rationals form a group under multiplication with identity element 1, and so do 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 a non-Abelian groupMathworldPlanetmath.

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

  • The complex numbers of absolute valueMathworldPlanetmathPlanetmathPlanetmath 1 form a group under multiplication, best thought of as the unit circleMathworldPlanetmath. The quaternions of absolute value 1 form a group under multiplication, best thought of as the three-dimensional unit sphereMathworldPlanetmath S3. The two-dimensional sphere S2 however is not a group in any natural way.

  • The positive integers less than n which are coprimeMathworldPlanetmathPlanetmath to n form a group if the operation is defined as multiplication modulo n. This is an Abelian group whose order is given by the Euler 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 into topological groups.

Symmetry groups

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 can define the fundamental groupMathworldPlanetmath of X at x. It consists of (homotopy classes of) continuousMathworldPlanetmathPlanetmath paths starting and ending at x and describes the structureMathworldPlanetmath of 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 among their 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 R forms 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 81053 elements.

Title examples of groups
Canonical name ExamplesOfGroups
Date of creation 2013-03-22 12:49:19
Last modified on 2013-03-22 12:49:19
Owner AxelBoldt (56)
Last modified by AxelBoldt (56)
Numerical id 34
Author AxelBoldt (56)
Entry type Example
Classification msc 20-00
Classification msc 20A05
Related topic ExamplesOfFiniteSimpleGroups
Related topic SpinGroup
Related topic ExamplesOfAlgebraicKTheoryGroups
Related topic QuantumGroups
Related topic GroupsOfSmallOrder
Related topic TriangleGroups