This is a useful approach i have seen used in some results. If any abelian group g has order a multiple of p, then g must contain an element of order p. 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. Browse other questions tagged abstractalgebra group theory finite groups tensor products abelian groups or ask your own question. Applications to second homology groups, topology 23, no. The nonabelian tensor product of finite groups is finite graham j. Most of the essential structural results of the theory follow immediately from the structure theory of semisimple algebras, and so this topic occupies a long chapter. For every subgroup hof gthere is a subgroup kof gwith hk g and h\k feg. Kishimoto it is shown that the number of extremal traces on the fixedpoint algebra 9 equals the cardinality of. We shall prove the fundamental theorem of finite abelian groups which tells us that every finite abelian group is isomorphic to a direct product of cyclic \p\groups. The nonabelian tensor squares of finite pgroups have been studied. Isomorphism between tensor products of abelian groups. This is the content of the fundamental theorem for finite abelian groups.
In 9 and 21, the authors prove that the non abelian tensor product of finite groups is a finite group, and they also show that the non abelian tensor product of finite p groups is a finite p. Representations of algebras and finite groups 7 preface these notes describe the basic ideas of the theory of representations of nite groups. For every abelian group g and every balanced product. Representation theory university of california, berkeley. A homologyfree proof article pdf available in glasgow mathematical journal 5203. Tensor product of finite groups is finite groupprops. Every nite abelian group a can be expressed as a direct sum of cyclic groups of primepower order. The theory of abelian groups is generally simpler than that of their non abelian counterparts, and finite abelian groups are very well understood. More precisely, if g is abelian and h is a nilpotent group, then an upper bound for the exponent of g. The role of the tensor product in the splitting of abelian groups. The fixedpoint algebra of tensorproduct actions of. Some of the general structure theory in the compact case is quite similar to that of the case of. Direct products and classification of finite abelian groups. After explicating a minimalist notion of reasonability, we will see that a tensor product a z q is just right.
Deligne tensor product of abelian categories in nlab. If x is of finite order, then yf is a torsion group and splits trivially. He argues that in this case it is better to use kellys tensor product of finitely cocomplete categories, because it always exists, and it agrees with delignes tensor product. Let g and h be groups that act compatibly on each other. Suppose that g is residually finite and the subgroup \g,h \langle g1gh \ \mid g \in g, h\in h\rangle \ satisfies some nontrivial identity \f \equiv 1\. Suppose and are abelian groups possibly equal, possibly distinct. A nonabelian tensor product of lie algebras glasgow. In fact, the claim is true if k 1 because any group of prime order is a cyclic group, and in this case any nonidentity element will. Every nite abelian group is isomorphic to a direct product of cyclic groups of orders that are powers of prime numbers.
Aata finite abelian groups university of puget sound. Ellis department of pure mathematics, university college of north wales, bangor, gwynedd ll57 zuw, united kingdom communicated by p. A bilinear map is a function of two variables that belong. Tensor products of affine and formal abelian groups arxiv. Direct products and classification of finite abelian. Finiteness of a non abelian tensor product of groups article pdf available in theory and applications of categories 25.
The non abelian tensor product of finite groups is finite by graham ellis, journal of algebra, issn 00218693, volume 111, page 203 205year 1987. This is a group via pointwise operations, so it is clearly abelian. Theorem let a be a finite abelian group of order n. The concept of a free group is discussed first in chapter 1 and in chapter 2 the tensor product of two groups for which we write a. Official copy elsevier sciencedirect, via doi more info. Nonabelian tensor product of residually finite groups.
Some remarks on history the earliest paper which uses a version of the nonabelian tensor square, and so a replacement of the commutator map by a morphism, is surely 1 by claire miller. We brie y discuss some consequences of this theorem, including the classi cation of nite. Kishimoto it is shown that the number of extremal traces on the fixedpoint algebra d g equals the cardinality of the subgroup k of automorphisms in g which are weakly inner in the trace representation of d. Let ab be the collection of abelian groups, regarded as a multicategory whose multimorphisms are the multilinear maps a 1. Pdf the nonabelian tensor product of finite groups is finite. As this paper is simply an introduction into the simplest forms of representation theory, we deal exclusively with nite groups, in both the abelian and non abelian case. While it is comparatively simple to do so for nite groups and there are known methods for doing so, it is often very di cult to do so for in nite groups. Statement from exam iii pgroups proof invariants theorem. Xi, 1960 tensor product of nonabelian groups and exact sequences 167 independent of the choice of presentation of c, but that it does contain tor c, g, where c and are c and g. Let g be a finite abelian group acting by tensorproduct automorphisms on a uhfcalgebra 8. Every finite abelian group is isomorphic to a product of cyclic groups of primepower orders. In case that both the groups are abelian groups and both the maps are trivial maps i.
Ii for tensor products they wrote \direct products of hilbert spaces. Finally, an abelian group is constructed by taking nonabelian tensor product of groups. There are many examples of application of the construction and universal properties of. We denote by \\eta g,h\ a certain extension of the nonabelian tensor product \g \otimes h\ by \g \times h\. The non abelian tensor product of finite groups is finite. Pairing, tensor product, finite abelian group, module, duality. A generalized tensor product of groups was introduced by r. Finiteness conditions for the nonabelian tensor product of. In this paper we study tensor products of affine abelian group. B c a \times b \to c that are right exact in each argument separately. Since and are abelian, is a normal subgroup of both and thus is also a normal subgroup of because an element of is a finite product of elements of and.
B is defined by factoring out an appropriate subgroup of the free group on the cartesian product of the two groups. For a a and b b two abelian categories, their deligne tensor product a. Pdf the nonabelian tensor product of finite groups is. We prove that if p is a prime and every tensor has p. Smith normal form is a reduced form similar to the row reduced matrices encountered in elementary linear algebra. The nonabelian tensor product of finite groups is finite, j. Their tensor product as abelian groups, denoted or simply as, is defined as their tensor product as modules over the ring of integers note that in case are abelian groups but are also being thought of as modules over some other ring for instance, as vector spaces over a field then. Let g, h be groups that act compatibly on each other and consider the non abelian tensor product \g \otimes h\. In 9 and 21, the authors prove that the nonabelian tensor product of finite groups is a finite group, and they also show that the nonabelian tensor product of finite p groups is a finite p. Direct products and classification of finite abelian groups 16a. For a and b two abelian groups, their tensor product a.
Tensor products of affine and formal abelian groups tilman bauer and magnus carlson abstract. The nonabelian tensor product of finite groups is finite by graham ellis, journal of algebra, issn 00218693, volume 111, page 203 205year 1987. Using our results, we obtain some upper bounds for the exponent of the schur multiplier of the nonabelian tensor product of groups. Direct products and finitely generated abelian groups note. In this paper we shall consider abelian groups only. Tensor products of modules over a commutative ring are due to bourbaki 2 in 1948. In the previous section, we took given groups and explored the existence of subgroups. Their tensor product as abelian groups, denoted or simply as, is defined as the quotient of the free abelian group on the set of all symbols by the following relations. The tensor product of nonabelian groups and exact sequences. For finite groups g and h the nonabelian tensor product of groups is finite 25. On the nonabelian tensor product of groups algebra colloquium. Let be a finite non abelian group in which every proper subgroup is abelian. Classify all representations of a given group g, up to isomorphism. 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.
The fixedpoint algebra of tensorproduct actions of finite. As with all universal properties, the above property defines the tensor product uniquely up to a unique isomorphism. Kishimoto it is shown that the number of extremal traces on the fixedpoint algebra 9. Pdf finiteness of a nonabelian tensor product of groups. B c a \boxtimes b \to c are equivalent to functors a. Prove the existence part of the fundamental structure theorem for abelian groups. On the tensor square of non abelian nilpotent finite. Xi, 1960 tensor product of non abelian groups and exact sequences 167 independent of the choice of presentation of c, but that it does contain tor c, g, where c and are c and g.
Pdf in this note, we give a homologyfree proof that the nonabelian tensor product of two finite groups is finite. And of course the product of the powers of orders of these cyclic groups is the order of the original group. 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. Pairings from a tensor product point of view archive ouverte hal. The nonabelian tensor product of finite groups is finite. Here the author points out that while delignes tensor product always exists for finite abelian categories, it does not always exist for general abelian categories. In this section, we introduce a process to build new bigger groups from known groups. B a, b \mapsto a \otimes b in this multicategory is the tensor product of abelian groups.
Tensors of free groups and abelian groups physics forums. Characters on nite abelian groups satisfy a sort of hahnbanach theorem. The fundamental thm of finite abelian groups we are now in a position to give a complete classification of all finite abelian groups. Note that the primes p 1p r are not necessarily distinct. The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules and vector spaces, are developed. These actions form a compatible pair of actions, hence it makes sense to take the tensor product of the two groups. Pointed finite tensor categories over abelian groups arxiv. B a \boxtimes b is the abelian category such that for any other abelian category c c right exact functors of the form a.
The tensor product of two abelian groups ubc library open. This should be a special case of the tensor product whose existence i need and i really need this tensor product not only for the discrete, torsion free case, but in general. 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. B is a new abelian group which is such that a group homomorphism a. Moreover the powers pe 1 1p er r are uniquely determined by a. The fundamental thm of finite abelian gps every finite abelian group is a direct product of cyclic groups of prime power order, uniquely determined up to the order in which the factors of the product are written. Let g be a finite abelian group acting by tensorproduct automorphisms on a uhfc.
Finite groups and character theory this semester well be studying representations of lie groups, mostly compact lie groups. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Combining the little we know about character groups with the fundamental theorem of nite abelian groups we get proposition 8. A nonabelian tensor product of lie algebras volume 33 issue 1 graham j.
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