Category of groups

Category of groups
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	Category of groups
Caption	Commutative diagram of group homomorphisms
Type	Category
Objects	Groups
Morphisms	Group homomorphisms

Category of groups

The Category of groups is the category whose objects are groups such as Symmetric group S_n, Alternating group A_n, Dihedral group D_n, Quaternion group Q_8 and whose morphisms are group homomorphisms like the inclusion of Cyclic group C_n into Multiplicative group of complex roots of unity or the projection from a Direct product of groups to a factor. As an algebraic category it connects classical examples such as General linear group GL_n(F), Special linear group SL_n(F), Heisenberg group, Torsion group, Free group F_n with structural constructions like Free product with amalgamation, Direct limit, Inverse limit and functorial operations like Abelianization and center.

Definition and basic properties

Objects are groups drawn from families including Finite simple group, Cyclic group C_p, Sporadic group Monster group, Lie group, p-group and Solvable group, while morphisms are homomorphisms such as the canonical map from a Free group F_n to a quotient by a normal subgroup or the determinant homomorphism det: General linear group GL_n(Z) → units group. The category is concrete over Set via the forgetful functor to Set and is complete, cocomplete and has a zero object when restricted to the trivial group; examples include products like Direct product of groups and coproducts like Free product of groups. Standard constructions and counterexamples link to objects such as Infinite cyclic group Z, Prüfer p-group, Symmetric group S_3, Alternating group A_5 and Burnside groups.

Morphisms and underlying functors

Morphisms include monomorphisms like inclusions of Sylow p-subgroups, epimorphisms such as quotient maps by a Normal subgroup, and isomorphisms exemplified by conjugation automorphisms in Dihedral group D_n or inner automorphisms of Simple groups. The forgetful functor U: Category of groups → Set and the left adjoint free group functor F: Set → Category of groups relate examples like Free group F_n, Presentation of a group, Nielsen–Schreier theorem and the Universal property. Functors to other algebraic categories include abelianization to Category of abelian groups and embedding into Category of monoids or restriction to Category of rings via group rings exemplified by Group ring Z[G].

Limits, colimits, and constructions

Products and equalizers yield limits, with finite examples such as product of Cyclic group C_m and Cyclic group C_n and inverse limit constructions producing profinite groups like Galois groups and p-adic integers viewed via inverse systems. Coproducts realized by free products link to Kurosh subgroup theorem and amalgamated products tie to Seifert–van Kampen theorem in algebraic topology and groups acting on Bass–Serre trees. Pushouts appear in HNN extensions and constructions related to One-relator groups and Amalgamated free product of groups. Direct and inverse limits connect to examples such as Prüfer p-group as a direct limit and profinite completions of Fundamental groups.

Adjunctions and monadicity

The free/forgetful adjunction F ⊣ U between Set and the Category of groups is central: classical free groups Free group F_S and universal properties exemplify this adjunction along with presentations arising from generators and relations. The adjunctions with Category of abelian groups via abelianization and inclusion produce left and right adjoints exemplified by the commutator subgroup quotient and inclusion functors, relating to derived functors like Homology of groups and Group cohomology. Monadicity results show the Category of groups is monadic over Set with algebras presented by signature and equational laws; examples involve Lawvere theory descriptions, varieties and comparisons with monadic descriptions of Lie algebras and Rings.

Subobjects, quotients, and exactness

Subobjects correspond to subgroups such as Normal subgroup, Characteristic subgroup, Sylow subgroup and include examples like center and Derived subgroup. Quotients by normal subgroups yield factor groups encountered in Jordan–Hölder theorem and simple quotients like Alternating group A_5. Exactness properties are expressed via short exact sequences 1 → N → G → Q → 1 appearing in extensions, semidirect products linking to Split extension, and group cohomology classes in H^2(G, A) classifying extensions. Notions of kernel and cokernel are tied to normal closures, and counterexamples to abelian exactness appear in nonabelian extensions and constructions like Baer correspondence.

Automorphism and endomorphism objects

Endomorphisms and automorphisms form groups and monoids such as the Automorphism group Aut(G), Inner automorphism group Inn(G), Outer automorphism group Out(G) and endomorphism monoid End(G). Important examples include automorphism groups of Free group F_n (Out(F_n)), automorphisms of Finite simple group Monster group, and Galois connections between automorphisms of field extensions and Galois groups. Functorial automorphism constructions link to Group actions on combinatorial objects like Cayley graphs, actions on Riemann surfaces and applications to classification problems involving Modular group and mapping class groups.