TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
Grp (or Gp) has the
class Class 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 differently f ...
of all
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
for objects and
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s for
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s. As such, it is a
concrete category In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. The study of this category is known as
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
.

Relation to other categories

There are two
forgetful functor In mathematics, 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 'before' mapping to the output. For an algebraic structure of a given signa ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the
Grothendieck groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 functorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
; this functor assigns to every set ''S'' the
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
on ''S.''

Categorical properties

The
monomorphism In the context of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, rin ...
s in Grp are precisely the
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
homomorphisms, the
epimorphism 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...
s are precisely the
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
homomorphisms, and the
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s are precisely the
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
while the category-theoretical coproduct in Grp is the
free product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of groups. The
zero object In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
s in Grp are the
trivial groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 math Mathematics (from Greek: ) includes the study of such topics as quantity ( number theory), structure (algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit= ...
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 groupsIn mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...
, Ab, is a
full subcategory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of Grp. Ab is an
abelian category In mathematics, an abelian category is a Category (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mo ...
, but Grp is not. Indeed, Grp isn't even an
additive category In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
''S''3 of order three to itself, $E=\operatorname\left(S_3,S_3\right)$, 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s. A
finite ringIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
with no zero divisors is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, but there is no field with ten elements because every
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
has for its order, the power of a prime.

Exact sequences

The notion of
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same ...
is meaningful in Grp, and some results from the theory of abelian categories, such as the , the five lemma, 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 a pair of morphisms called kernel pairs, satisfying certain ''exactness'' conditions. In that way, regular categories recapture many ...
.

References

* {{DEFAULTSORT:Category Of Groups
Groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
Group theory