This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Notes Math. Google Scholar. Algebra15No. Algebra17No. Algebra 25No. Albrecht and T. Algebra21No. Albrecht, T. Faticoni, and H. Albrecht and H. Albrecht, H. Goeters, T. Faticoni, and W. Goeters, and W. Albrecht and J. Arnold and J. Algebra18No. Arnold and E. Arnold and C. Arnold, R. Pierce, J. Reid, C. Vinsonhaler, and W.
Algebra71No.Set [Domaine]. Abelian group n. Elliptic curve Linear algebraic group Abelian variety. In abstract algebraan abelian groupalso called a commutative groupis a group in which the result of applying the group operation to two group elements does not depend on their order the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers.
They are named after Niels Henrik Abel. The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebrawith many other basic objects, such as a module and a vector spacebeing its refinements. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood. On the other hand, the theory of infinite abelian groups is an area of current research.
More compactly, an abelian group is a commutative group. A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group". Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules.
The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered.
To verify that a finite group is abelian, a table matrix - known as a Cayley table - can be constructed in a similar fashion to a multiplication table. The group is abelian if and only if this table is symmetric about the main diagonal.
This implies that the ij 'th entry of the table equals the ji 'th entry, thus the table is symmetric about the main diagonal. In general, matriceseven invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative.
However, some groups of matrices are abelian groups under matrix multiplication - one example is the group of 2x2 rotation matrices. Abelian groups were named for Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of an equation implies its roots are solvable by radicals.
See Section 6. In this way, G becomes a module over the ring Z of integers. In fact, the modules over Z can be identified with the abelian groups. Theorems about abelian groups i. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain.
In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. This is not true if H is a non-abelian group. The set Hom GH of all group homomorphisms from G to H thus turns into an abelian group in its own right. Somewhat akin to the dimension of vector spacesevery abelian group has a rank. It is defined as the cardinality of the largest set of linearly independent elements of the group.Professionally, very few things meant more to me than his ongoing support and encouragement.
Advertisement Hide. Chapter First Online: 04 June This is a preview of subscription content, log in to check access. Balof, P. Keef, Invariants on primary abelian groups and a problem of Nunke. Note Mat. Eklof, A. Fuchs, Infinite Abelian Groupsvol. Fuchs, P. Hill, The balanced-projective dimension of abelian p -groups.
Keef, On the Tor functor and some classes of abelian groups. Contemporary Mathematicsvol. Keef, On iterated torsion products of abelian p -groups. Rocky Mountain J. Keef, Mahlo cardinals and the torsion product of primary abelian groups. Keef, Perfectly bounded classes of primary abelian groups.
Algebra Appl. Nunke, On the structure of Tor II. Keef 1 Email author 1. Personalised recommendations.Apple card we are unable to verify your identification information
Cite chapter How to cite? ENW EndNote. Buy options.In the present, third survey of reviews of articles on Abelian groups are included papers reviewed during — Here, as in the previous surveys, the questions concerning finite Abelian groups, topological groups, ordered groups, group algebras, modules, structures of subgroups, as well as the questions connected with logic, are not touched on.
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MathOverflow is a question and answer site for professional mathematicians.
It only takes a minute to sign up. I'm interested in all subgroups, not just up-to-isomorphism. These three answers were originally comments. I am answering the part of the question which was deleted:. Is there anything else interesting to say about the collection of subgroups of an [finite] abelian group. The answer by Amritanshu Prasad, comments by Derek Holt and Robin Chapman here provide much more very interesting information about enumerating subgroups.
Elementary theories of lattices of subgroups, Algebra i Logika 9 — and references there. This is a better reference than Taiclin.
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Algebra i Logika 14no. Update In fact, in the paper, Sapir, M. Varieties with a finite number of subquasivarieties. The problem of enumerating subgroups of a finite abelian group is both non-trivial and interesting.Module 17 - Fundamental Theorem of Finite Abelian Groups
It is also worthwhile to study this set as a lattice. As far as I know, the reference "Subgroup lattices and symmetric functions" by Lynne M. Butler Memoirs of the AMS, no. It is beautifully written and has a good survey of the history of the problem. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Subgroups of a finite abelian group Ask Question.
Asked 9 years, 11 months ago. Active 7 months ago. Viewed 5k times. BTW, this is not the right medium for questions like that. Then specialise this to the abelian case. Some of the above comments may disappear if they're no longer relevant i.
If you want to read them, you can find them at tea. Active Oldest Votes. I am answering the part of the question which was deleted: Is there anything else interesting to say about the collection of subgroups of an [finite] abelian group.Abelian Group Theory pp Cite as. Unable to display preview. Download preview PDF. Skip to main content.
Google Scholar. Arnold and E.
An Invariant on Primary Abelian Groups with Applications to Their Projective Dimensions
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Bass, K-theory and stable algebra, Pub. Crawley, The cancellation of torsion abelian groups in direct sums, J. Crawley and A. Hales, The structure of Abelian p-groups given by certain presentations, J. Evans, Jr. Fuchs, On a substitution property for modules, Monatshefte fur Math. Fuchs, Infinite Abelian Groups.
Goodearl, Power cancellation of groups and modules, Pacific J.Ac compressor clutch not engaging
Griffith, A solution to the splitting mixed group problem of Baer, Trans. Harrison, Infinite Abelian groups and homological methods, Ann.
Hunter, Balanced subgroups of abelian groups, Trans. Hunter, F. Richman, and E.This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Zametki, 7No. Google Scholar. Zavedenii, Matematika, No. Nauk Ukrain. SSR, Ser.Ki korle bacha hoy
A, No. Kuz'minov and I. Lapin and N. Mishina and L. SSR, No. Carolinae, 7No. Carolinae, 8No. Nauk, 25No.Token mono q
Young Moldavian Scientists, Phys. Nauk, Vladivostokpp. Abian and D. Palermo, 12No. Number Theory, 2No. Beaumont and R. Benabdallah, B. Eisenstadt, J. Irwin, and E. Benabdallah and J. Boyer and A.
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