Category of groups

In mathematics, the Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a . The study of this category is known as group theory. There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to . M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group 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 on S. The monomorphisms in Grp are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms. The category Grp is both . The in Grp is just the direct product of groups while the category-theoretical coproduct in Grp is the free product of groups. The zero objects in Grp are the trivial groups (consisting of just an identity element). Every morphism f : G → H in Grp has a (given by the ordinary kernel of algebra ker f = {x in G | f(x) = e}), and also a (given by the factor group 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. The , Ab, is a of Grp. Ab is an , but Grp is not. Indeed, Grp isn't even an , 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 S3 of order three to itself, , 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.

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Cokernel

The cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f. Cokernels are to the , hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain).

Trivial group

In mathematics, a trivial group or zero group is a group consisting 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 usually denoted as such: or depending on the context. If the group operation is denoted then it is defined by The similarly defined is also a group since its only element is its own inverse, and is hence the same as the trivial group.

Normal closure (group theory)

In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing Formally, if is a group and is a subset of the normal closure of is the intersection of all normal subgroups of containing : The normal closure is the smallest normal subgroup of containing in the sense that is a subset of every normal subgroup of that contains The subgroup is generated by the set of all conjugates of elements of in Therefore one can also write Any normal subgroup is equal to its normal

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