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In mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
This picture illustrates how the hours on a clock form a group under modular addition. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In topology, a continuous function is generally defined as one for which preimages of open sets are open. ...
Definition
Formally, we start with a category C with finite products (i.e. C has a terminal object 1 and any two objects of C have a product). A group object in C is an object G of C together with morphisms In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ...
In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C...
- m : G × G → G (thought of as the "group multiplication")
- e : 1 → G (thought of as the "inclusion of the identity element")
- inv: G → G (thought of as the "inversion operation")
such that the following properties (modeled on the group axioms) are satisfied - m is associative, i.e. m(m × idG) = m (idG × m) as morphisms G × G × G → G; here we identify G × (G × G) in a canonical manner with (G × G) × G.
- e is a two-sided unit of m, i.e. m (idG × e) = p1, where p1 : G × 1 → G is the canonical projection, and m (e × idG) = p2, where p2 : 1 × G → G is the canonical projection
- inv is a two-sided inverse for m, i.e. if d : G → G × G is the diagonal map, and eG : G → G is the composition of the unique morphism G → 1 (also called the counit) with e, then m (idG × inv) d = eG and m (inv × idG) d = eG.
Examples - A group can be viewed as a group object in the category of sets. The map m is the group operation, the map e (whose domain is a singleton) picks out the identity element of the group, and the map inv assigns to every group element its inverse. eG : G → G is the map that sends every element of G to the identity element.
- A topological group is a group object in the category of topological spaces with continuous functions.
- A Lie group is a group object in the category of smooth manifolds with smooth maps.
- A Lie supergroup is a group object in the category of supermanifolds.
- An algebraic group is a group object in the category of algebraic varieties. In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes.
- A localic group is a group object in the category of locales.
- The group objects in the category of groups (or monoids) are essentially the Abelian groups. The reason for this is that, if inv is assumed to be a homomorphism, then G must be abelian. More precisely: if A is an abelian group and we denote by m the group multiplication of A, by e the inclusion of the identity element, and by inv the inversion operation on A, then (A,m,e,inv) is a group object in the category of groups (or monoids). Conversely, if (A,m,e,inv) is a group object in one of those categories, then m necessarily coincides with the given operation on A, e is the inclusion of the given identity element on A, inv is the inversion operation and A with the given operation is an abelian group. See also Eckmann-Hilton argument.
This picture illustrates how the hours on a clock form a group under modular addition. ...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
In mathematics, a singleton is a set with exactly one element. ...
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...
A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...
On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...
In mathematics, a smooth function is one that is infinitely differentiable, i. ...
The concept of supergroup is a generalization of a that of group. ...
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. ...
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ...
In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ...
In mathematics, a group scheme is a group object (some would prefer to say just group) in the category of schemes. ...
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. ...
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesnt matter. ...
In mathematics, the Eckmann-Hilton argument (or Eckmann-Hilton principle or Eckmann-Hilton theorem) is an argument about monoid structures on a set where one is a homomorphism for the other. ...
Group theory generalized Much of group theory can be formulated in the context of the more general group objects. The notions of group homomorphism, subgroup, normal subgroup and the isomorphism theorems are typical examples. However, results of group theory that talk about individual elements, or the order of specific elements or subgroups, normally cannot be generalized to group objects in a straight-forward manner. Group theory is that branch of mathematics concerned with the study of groups. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ...
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