Oct 25, 2010 i have a homework problem which asks to prove that the subgroups of a finitely generated abelian group are finitely generated. Subgroup of finitely generated abelian group is finitely. All results are special cases of nitely generated modules over a principal ideal domain to be discussed in section iv. Geodesic lines in finitely generated abelian groups eventhoughtheorem1. Every nitely generated abelian group gis isomorphic to a direct sum of cyclic groups. The smallest subgroup containing xis the subgroup generated by x, denoted hxi. Z where the p i are primes, not necessarily distinct, and. This is the fundamental theorem of finitely generated abelian groups.
We brie y discuss some consequences of this theorem, including the classi cation of nite. The group of rational numbers is not finitely generated. We already know a lot of nitely generated abelian groups, namely cyclic groups, and we know they are all isomorphic to z n if they are nite and the only in nite cyclic group is z, up to isomorphism. A group is finitely generated if it has a finite set of generators. I have a homework problem which asks to prove that the subgroups of a finitely generated abelian group are finitely generated. F, where t is the torsion subgroup and f is a free abelian group. The set of complex and real numbers are both uncountable, and any. Jonathan pakianathan november 1, 2003 1 direct products and direct sums we discuss two universal constructions in this section. Every finitely generated abelian group is a direct sum of cyclic groups, that is, of the form. In general it is not easy to decide whether the intersections of nitely generated subgroups of metabelian groups as a whole are nitely generated.
The fundamental theorem of finitely generated abelian groups just for fun, here is the classi cation theorem for all nitely generated abelian groups. A primary cyclic group is one whose order is a power of a prime. In this section we prove the fundamental theorem of finitely generated abelian groups. I am interested if there is an example of an infinite finitely generated nonamenable group that is residually finite but does not contain nonabelian free subgroups.
The structure theorem for finitely generated abelian groups. Structure theorem for finitely generated abelian groups. Classification theorem for finitely generated abelian groups. By this we mean we can draw up a list albeit infinite of standard examples, no two of which are isomorphic, so that if we are presented with an arbitrary finitely generated abelian group, it is isomorphic to one on our list. Fundamental theorem of finitely generated abelian groups and its application in this post, we study the fundamental theorem of finitely generated abelian groups, and as an application. In this section, we introduce a process to build new bigger groups from known groups. Finitely generated abelian groups we begin this lecture by proving that the cyclic group of order nm is isomorphic to the direct product of cyclic groups of.
Fundamental theorem of finitely generated abelian groups and. 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. The fundamental theorem implies that every nite abelian group can be written up to isomorphism in the form z p 1 1 z p 2 2 z n n. On abelian subgroups of finitely generated metabelian groups. Definition 1 a group g is nitely generated if there is a nite subset a g such that g. Finitely generated abelian groups math 4120, modern algebra 6 7. If any abelian group g has order a multiple of p, then g must contain an element of order p. Which finitely generated abelian groups admit isomorphic. Modern algebra abstract algebra made easypart 7direct. 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. At first sight, finitely generated abelian groups and canonical forms of matrices appear to have little in common.
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. Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The structure theorem for finitely generated abelian groups states the following things. The most relevant part of the paper is the classification of the finitely generated abelian groups of units realizable with torsion. Every finitely generated abelian group can be expressed as the direct product of finitely many cyclic groups in other words, it is isomorphic to the external direct product of finitely many cyclic groups. Finitely generated groups university of california, berkeley. However, not all in nite subsets in an arbitrary nitely generated abelian group or even a free abelian group of nite. The structure theorem for finitely generated abelian groups mark cerenzia 29 july 2009 abstract this paper provides a thorough explication of the structure theorem for abelian groups and of the background information necessary to prove it. Pdf normal subgroups of glnd are not finitely generated. Introduction this paper gives some sufficient conditions for the finite presentability of nilpotentbyabelian groups. The finite abelian group is just the torsion subgroup of g. If are all finitely generated groups, so is the external direct product. Is there a slick proof of the classification of finitely.
The first summands are the torsion subgroup, and the last one is the free subgroup. The fundamental theorem of finite abelian groups every nite abelian group is isomorphic to a direct product of cyclic groups of prime power order. Finitely generated abelian groups g t abelian group additive notation i given e g ne z x x ux t cn times if u o c e it n o y k t c k c u times 17 uco carefall. However, it turns out to be not much more difficult to find the structure of a wider class of abelian groups, namely those which are generated by a finite subset of their elements. We prove the additive group of rational numbers is not finitely generated, and the multiplicative group of nonzero rational numbers is not finitely generated. Groves university of melbourne, parkville, australia 3052 communicated by peter h. In the previous section, we took given groups and explored the existence of subgroups. To explain this, we shall use the following result about the identity functor on the category of abelian groups or of nitely generated abelian groups. The classification of nonfinitely generated abelian groups is an open problem. See status of the classification of nonfinitely generated abelian groups. If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups. Pdf on abelian subgroups of finitely generated metabelian. If is a finitely generated group and is a normal subgroup of, then the quotient group is a finitely generated.
Then there exist a nonnegative integer t and if t 0 integers 1 finitely generated abelian groups math 4120, modern algebra 6 7 the fundamental theorem of finitely generated abelian groups just for fun, here is the classi cation theorem for all nitely generated abelian. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. Then there exist a nonnegative integer t and if t 0 integers 1 d are not finitely generated article pdf available in proceedings of the american mathematical society 1286 january 2000 with 88 reads how we measure reads. Some finitely presented nilpotent byabelian groups j. The rank of g is defined as the rank of the torsionfree part of g. Subgroups of a finitely generated abelian group physics. Infinite finitely generated nonamenable groups mathoverflow. However, the splitting of aas a direct sum is not natural. Finitely generated abelian groups may be classified. The classification of non finitely generated abelian groups is an open problem. Aug, 2012 in this lecture, i define and explain in detail what finitely generated abelian groups are. Theorem fundamental theorem of finitely generated abelian groups. The alternating group a 4 is an example of a finite solvable group that is not supersolvable. 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.
I be a collection of groups indexed by an index set i. If is a finitely generated group and is a subgroup of finite index in, then is also a finitely generated group. That is, every finitely generated abelian group is isomorphic to a group of the form. However, reduction to smith normal form, named after its originator h. However, it turns out to be not much more difficult to find the structure of a wider class of abelian groups, namely those which are generated by a finite subset of their.
In this section, all groups are abelian, and we use additive notation. Fundamental theorem of finitely generated abelian groups. In this lecture, i define and explain in detail what finitely generated abelian groups are. Here we shall prove that there are algorithms which decide for a free metabelian group or, more generally, for the wreath product of two free abelian groups whether the intersection of. A finitely generated abelian group is free if and only if it is torsionfree, that is, it contains no element of finite order other than the identity. A note on nielsen equivalence in finitely generated abelian groups volume 84 issue 1 daniel oancea. Find all abelian groups, up to isomorphism, of order 720. Finitelygenerated abelian groups structure theorem. Normal subgroups of gln d are not finitely generated article pdf available in proceedings of the american mathematical society 1286 january 2000 with 88 reads how we measure reads. We rst consider some theorems related to abelian groups and to r. Finitelygenerated abelian groups structure theorem for.
Subgroups of a finitely generated abelian group physics forums. Smith in 1861, is a matrix version of the euclidean algorithm and is exactly what the theory requires in both. We remark that the study of the group of units of torsion. 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. Direct products and finitely generated abelian groups note. I give examples, proofs, and some interesting tidbits that are hard to come by. The classification theorem for finitely generated abelian. Finitely generated abelian groups of units del corso. Smith normal form is a reduced form similar to the row reduced matrices encountered in elementary linear algebra. Available formats pdf please select a format to send. Finitely generated abelian groups and similarity of. The hint in the book says to prove it by induction on the size of x where the group g. Finitelygenerated abelian groups structure theorem for finitelygenerated abelian groups.
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