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[parent] examples of groups (Example)

Groups are ubiquitous throughout mathematics. Many “naturally occurring” groups are either groups of numbers (typically Abelian) or groups of symmetries (typically non-Abelian).

Groups of numbers

Most groups of numbers carry natural topologies turning them into topological groups.

Symmetry groups

Algebraic groups

Examples of Algebraic K-Theory groups

Quantum groups and Spin 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 group of $ X$ at $ x$. It consists of (homotopy classes of) continuous paths starting and ending at $ x$ and describes the structure of the “holes” in $ X$ accessible from $ x$. The fundamental group is generalized by the higher homotopy groups.
  • Other groups studied in algebraic topology are the homology groups of a topological space. In a different way, they also provide information about the “holes” of the space.
  • The free groups are important in algebraic topology. In a sense, they are the most general groups, having only those relations 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 inverses; the operation comes from string concatenation.
  • If $ A$ and $ B$ are two Abelian groups (or modules over the same ring), then the set $ \operatorname{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 $ {\cal P}(X)$ of $ X$ becomes an abelian group if we use the symmetric difference as operation. In this group, any element is its own inverse, which makes it into a vector space over $ \Bbb{Z}_2$.
  • If $ R$ is a ring with multiplicative identity, then the set of all invertible $ n\times n$ matrices over $ R$ forms a group under matrix multiplication with the identity matrix as identity element; this group is denoted by $ \operatorname{GL}(n,R)$. It is the group of units of the ring of all $ n\times n$ matrices over $ R$. For a given $ n$, the groups $ \operatorname{GL}(n,R)$ with commutative ring $ R$ can be viewed as the points on the general linear group scheme $ \operatorname{GL}_n$.
  • If $ K$ is a number field, then multiplication of (equivalence classes of) non-zero ideals in the ring of algebraic integers $ \cal{O}_K$ gives rise to the ideal class group of $ K$.
  • 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)\in K^2\mid y^2=x^3-x\}$, where $ K$ is any field. Every straight line intersects this set in three points (counting a point twice if the line is tangent, 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 varieties.
  • Let $ E$ be an elliptic curve 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 generated abelian group. The curve $ C$ in the example above is an elliptic curve defined over $ \mathbb{Q}$, thus $ C(\mathbb{Q})$ 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 $ 8\cdot 10^{53} $ elements.



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See Also: examples of finite simple groups, spin groups, examples of algebraic K-theory groups, quantum groups, groups of small order, triangle groups


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Cross-references: simple groups, finitely generated, elliptic curve, abelian varieties, infinity, tangent, intersects, line, straight, multiplicative functions, Dirichlet convolution, arithmetic functions, ideal class group, algebraic integers, ideals, equivalence classes, number field, general linear group scheme, commutative ring, group of units, identity matrix, vector space, symmetric difference, powerset, sum, commutativity, modules, concatenation, inverses, strings, finite, axioms, free groups, homology groups, higher homotopy groups, paths, continuous, classes, homotopy, fundamental group, point, groups of small order, Klein 4-group, smooth maps, group operations, differentiable manifolds, Lie groups, determinant, special orthogonal group, orthogonal, orthogonal group, combinations, transformations, dilations, contains, matrix multiplication, matrices, general linear group, matrix groups, rational numbers, algebraic extension, inverse Galois problem, Galois groups, equations, polynomial, solutions, field extensions, Galois theory, automorphism group, homomorphisms, composition, category, topological space, graph, automorphisms, dihedral group, square, cone, reflections, rotations, object, examples of finite simple groups, simplicity of the alternating groups, proof, simple, index, normal, transpositions, even number, product, even, alternating group, Cayley theorem, subgroup, finite group, permutations, symmetric group, topological groups, topologies, examples of rings, ring multiplication, invertible, multiplicative identity, units of quadratic fields, negatives, powers, ring, units, Euler, order, abelian group, coprime, sphere, unit sphere, unit circle, absolute value, multiplicative group, additive group, field, non-abelian group, quaternions, complex numbers, algebraic numbers, multiplication, positive, complex, real, rational, isomorphic, cyclic group, identity element, operation, addition, integers, non-Abelian, symmetries, abelian, numbers
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This is version 28 of examples of groups, born on 2002-06-28, modified 2008-09-21.
Object id is 3144, canonical name is ExamplesOfGroups.
Accessed 14115 times total.

Classification:
AMS MSC20-00 (Group theory and generalizations :: General reference works )
 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)

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