mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the category Grp (or Gp) has the

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

of all groups for objects and

group homomorphism In mathematics, 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) \cdot h(v) whe ...

s for morphisms. As such, it is a

concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects of the category as sets with additional ...

. The study of this category is known as

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

Relation to other categories

There are two

forgetful functor In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure or properties mapping to the output. For an algebraic structure of ...

s from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to sets. M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the

Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a group homomorp ...

of that monoid. The forgetful functor U: Grp → Set has a left adjoint given by the composite KF: Set→Mon→Grp, where F is the free functor; this functor assigns to every set ''S'' 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''− ...

on ''S.''

Categorical properties

The

monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphis ...

s in Grp are precisely the

injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...

homomorphisms, the

epimorphism In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analo ...

s are precisely the

surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

homomorphisms, and the

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

s are precisely the

bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

homomorphisms. The category Grp is both complete and co-complete. The category-theoretical product in Grp is just the

direct product of groups In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is o ...

while the category-theoretical coproduct in Grp is the

free product In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...

of groups. The

zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

s in Grp are the

trivial group In mathematics, a trivial group or zero group is a group that consists of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usu ...

s (consisting of just an identity element). Every morphism ''f'' : ''G'' → ''H'' in Grp has a category-theoretic kernel (given by the ordinary kernel of algebra ker f = ), and also a category-theoretic cokernel (given by the

factor group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For exam ...

of ''H'' by the normal closure of ''f''(''G'') in ''H''). Unlike in abelian categories, it is not true that every monomorphism in Grp is the kernel of its cokernel.

Not additive and therefore not abelian

The

category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object o ...

, Ab, is a

full subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...

of Grp. Ab is an

abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category o ...

, but Grp is not. Indeed, Grp isn't even an

additive category In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. Definition There are two equivalent definitions of an additive category: One as a category equipped wit ...

, because there is no natural way to define the "sum" of two group homomorphisms. A proof of this is as follows: The set of morphisms from 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 grou ...

''S''₃ of order three to itself,

E=\operatorname(S_3,S_3)

, has ten elements: an element ''z'' whose product on either side with every element of ''E'' is ''z'' (the homomorphism sending every element to the identity), three elements such that their product on one fixed side is always itself (the projections onto the three subgroups of order two), and six automorphisms. If Grp were an additive category, then this set ''E'' of ten elements would be a ring. In any ring, the zero element is singled out by the property that 0''x''=''x''0=0 for all ''x'' in the ring, and so ''z'' would have to be the zero of ''E''. However, there are no two nonzero elements of ''E'' whose product is ''z'', so this finite ring would have no

zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...

s. A finite ring with no zero divisors is a field by Wedderburn's little theorem, but there is no field with ten elements because every

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

has for its order, the power of a prime.

Exact sequences

The notion of

exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definit ...

is meaningful in Grp, and some results from the theory of abelian categories, such as the nine lemma, the

five lemma In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma (mathematics), lemma about commutative diagrams. The five lemma is not only valid for abelian cat ...

, and their consequences hold true in Grp. Grp is a

regular category In category theory, a regular category is a category with limit (category theory), finite limits and coequalizers of all pairs of morphisms called kernel pairs, satisfying certain ''exactness'' conditions. In that way, regular categories recapture ...

References

* {{DEFAULTSORT:Category Of Groups Groups Group theory