We also establish a sufficient condition for a finitely generated non abelian tensor product to be finite. Tensor products of finitely cocomplete and abelian categories. Resource d finitelygenerated abelian groups mordells theorem tells us that the rational points on an elliptic curve eq form a nitelygenerated abelian group. Conversely, given nitely many prime powers, arrange them in a table such as. The qtensor square of finitely generated nilpotent groups, q. Every abelian category a is a module over the monoidal category of finitely generated abelian groups. Finiteness conditions for the nonabelian tensor product.
Given an abelian group a, ta is the abelian group with generators ya, aea, and defining relations ha1 74 yabc ya yb yc dab ybc ycax 12 for all a, 6, c e a. Groves department of mathematics, university of melbourne, parkville, australia communicated by graham higman received september 20, 1983 1. Prove that the direct product of abelian groups is abelian. Further, we show that if n is a positive integer and every tensor is left nengel in \\eta g,h\, then the non abelian tensor product \g \otimes h\ is locally nilpotent. Every nite abelian group is isomorphic to a direct product of cyclic groups of orders that are powers of prime numbers. Direct products and finitely generated abelian groups note. For now, see properties for the more general discussion. Classi cation of finitely generated abelian groups the proof given below uses vector space techniques smith normal form and generalizes from abelian groups to \modules over pids essentially generalized vector spaces. Tensor product of two finitelygenerated modules over a local ring is zero. If we combine this with the other properties in theorem 2, we can already calculate all tensor products m. The following properties are not hard to check, and allow computation of i in the finitely generated case 151. The tensor product of two abelian groups ubc library open.
Our method for computing the nonabelian tensor squares of will rely on the fact that this group is polycyclic. The fundamental theorem of finite abelian groups wolfram. Nonabelian tensor product of residually finite groups. Structure of finitely generated abelian groups abstract the fundamental theorem of finitely generated abelian groups describes precisely what its name suggests, a fundamental structure underlying finitely generated abelian groups. We brie y discuss some consequences of this theorem, including the classi cation of nite. Is this tensor product of two modules finitely generated. We recall the fundamental theorem of finitely generated abelian groups. Direct products and classification of finite abelian groups 16a. In particular, taking r to be this shows every tensor product of modules is a quotient of a tensor product of abelian groups. Let m be a finitely generated rmodule generated by a minimal generating. G h is an isomorphism then the restrictions of f to gt and gp respectively are isomorphisms giving.
Finiteness conditions for the nonabelian tensor product of. Every finitely generated abelian group is a direct sum of cyclic groups, that is, of the form. Note that trivial mutual actions form a compatible pair of actions, so the definition applies. Click download or read online button to get abelian categories with applications to rings and modules book now. Further, we show that if n is a positive integer and every tensor is left nengel in \\eta g,h\, then the nonabelian tensor. Is there a slick proof of the classification of finitely. Let k be a field, and let a be a finitely generated kalgebra. The tensor product of abelian groups agrees with the tensor product of groups if we assume both groups to act trivially on each other.
Finitelygenerated abelian groups structure theorem. Finitelygenerated abelian groups 5 thus as a whole, the torsion subgroup takes the form of a product of primepower cyclic groups, g tor. Th freee abelian group over a set the subjec otf this paper, the tensor product of two abelian groups, involve intrinsicalls thye concept of a free group th tensoe r product m our case might better be termed the free bilinear product, as by fuchs and so this topic wil l be dealt with first in its own right. Finiteness of a nonabelian tensor product of groups nick inassaridze transmitted by ronald brown abstract. Tensor powers of modules over finitely generated abelian. Feb 25, 2017 the direct product is a way to combine two groups into a new, larger group. We provide necessary and sufficient conditions for the non abelian tensor product of finitely generated groups to be finitely generated. Abelian categories with applications to rings and modules. Pdf the order of the nonabelian tensor product of groups. Pdf the nonabelian tensor product of finite groups is finite. The tensor product of finitely generated groups in this chapter we shall determine the structure of the tensor product of two finitely generated groups.
Smith normal form is a reduced form similar to the row reduced matrices encountered in elementary linear algebra. Jonathan pakianathan november 1, 2003 1 direct products and direct sums we discuss two universal constructions in this section. For these reasons it is important to know the structure of abelian groups. Some computations of nonabelian tensor products of groups. The fundamental theorem of finite abelian groups states that a finite abelian group is isomorphic to a direct product of cyclic groups of primepower order, where the decomposition is unique up to the order in which the factors are written. The following relates the tensor product to bilinear functions.
Finite abelian groups amin witno abstract we detail the proof of the fundamental theorem of nite abelian groups, which states that every nite abelian group is isomorphic to the direct product of a unique collection of cyclic groups of prime power orders. Some finiteness conditions for g in terms of certain torsion elements of the non abelian tensor square \g \otimes g\ are also studied. Structure of finitely generated modules over a pid. Tensor products rst arose for vector spaces, and this is the only setting where tensor products occur in physics and engineering, so well describe the tensor product of vector. Cancelling finitely generated projective modules from a tensor product of finitely generated projective modules. The tensor product of two abelian groups by david mit ton a.
Let g, h be groups that act compatibly on each other and consider the nonabelian tensor product g. Nonabelian tensor products of solvable groups request pdf. In 27 the second author derived some properties of the non abelian tensor square of a group g via its embedding in a larger group. It is a definition or a proposition dependening on whether one takes the notion of bilinear function to be defined before or after that of tensor product of abelian groups. The primary decomposition formulation states that every finitely generated abelian group g is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups.
Tensor product of two finitely generated modules over a local ring is zero. We also establish a sufficient condition for a finitely generated nonabelian tensor product to be finite. A primary cyclic group is one whose order is a power of a prime. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. Classification theorem for finitely generated abelian groups. Then a c b is an abelian group, namely the tensor product of a and b as abelian groups. Direct products and classification of finite abelian groups. Aug 18, 2017 we prove that if p is a prime and every tensor has ppower order, then the non abelian tensor product \g \otimes h\ is locally finite. Note that we have associated with each object nin the category of amodules an object hom am. Finally, the universal property which characterizes the tensor product of abelian groups follows for a c b from its very definition. In case that both the groups are abelian groups and both the maps are trivial maps i. Statement from exam iii pgroups proof invariants theorem.
In the previous section, we took given groups and explored the existence of subgroups. A generalised tensor product g 0 h of groups g, h has been introduced by r. Since the polycyclics property is closed under extensions, we have is polycyclic. Z n for finitely generated abelian groups m and n, or. We prove that central extensions of most finite simple.
I gi is an isomorphism, then the restrictions of g to mg and gm respectively are isomorphisms giving mg. 1 we explore the structure of the functor from kalgebras to abelian groups given by b. And of course the product of the powers of orders of these cyclic groups is the order of the original group. Finitelygenerated abelian groups structure theorem for. In this section, we introduce a process to build new bigger groups from known groups. By lemma 1 we see that is an extension of a finitely generated abelian group by a finite cyclic group. The dual of a finitely generated module is reflexive, that is, isomorphic to its own double dual. The main problem is that tensor product can create torsion and cotorsion and reflexive modules have neither. In other words, a left rmodule is the same as an abelian group. This is the fundamental theorem of finitely generated abelian groups. The reason why g 0 h does not necessarily reduce to guh oz huh, the usual tensor product. The first summands are the torsion subgroup, and the last one is the free subgroup. Computation in a direct product of n groups consists of computing using the individual group operations in each of the n components. These actions form a compatible pair of actions, hence it makes sense to take the tensor product of the two groups.
As an application of the propositions so far, we calculate the the tensor product of two. The concept of a direct sum of abelian groups wi l l be used extensively and therefore a workable definition wi l l f i rs t be formulated. The classification theorem for finitely generated abelian. Cohomology of the group ring of finitely generated abelian groups andr. Every free abelian group may be described as a direct sum of copies of, with one copy for each member of its basis. Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian. In a direct product of abelian groups, the individual group operations are all commutative, and it follows at once that the direct product is an. R n is an abelian group, constructed as a quotient group.
We can also consider the tensor product as a functor m. N in the category of abelian groups and with each homomorphismf. Just as you can factor integers into prime numbers, you can break apart some groups into a direct product of simpler groups. The purpose of this article is to study the existence of deligne. Finitely generated abelian groups we discuss the fundamental theorem of abelian groups to give a concrete illus. A bvalgebra structure on hochschild cohomology of the group. That is, every finitely generated abelian group is isomorphic to a group of the form. The tensor product of two free abelian groups is always free abelian, with a basis that is the cartesian product of the bases for the two groups in the product. Non abelian tensor products of groups have been studied by a number of authors. The existence of algorithms for smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums. We prove that if p is a prime and every tensor has ppower order, then the nonabelian tensor product \g \otimes h\ is locally finite. He argues that in this case it is better to use kellys tensor product of finitely.
With abelian groups, additive notation is often used instead of multiplicative notation. We will use a to denote the algebra rcannm of endomorphisms of m induced by rg and i. In this section we prove the fundamental theorem of finitely generated abelian groups. Unless otherwise indicated, all tensor products are assumed to be over r.
As in this introduction, r will denote a commutative ring with unity, g a finitely generated abelian group and m a finitely generated tmodule. Since for finitely generated abelian groups cohomology. The present paper is concerned solely with this generalized tensor product. Direct products of groups abstract algebra youtube. Some finiteness conditions for g in terms of certain torsion elements of the nonabelian tensor square \g \otimes g\ are also studied. It arises in applications in homotopy theory of a generalised van kampen theorem. We provide necessary and sufficient conditions for the nonabelian tensor product of finitely generated groups to be finitely generated. The finitely generated abelian groups, together with the group homomorphisms, form an abelian category which is a serre subcategory of the category of abelian groups. This site is like a library, use search box in the. Deligne tensor product of abelian categories in nlab. Any finitely generated abelian group is polycyclic. We will represent finitely generated abelian groups via generators and relations.
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