Abelian group

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Commutative group (mathematics)

For the group described by the archaic use of the related term "Abelian linear group", see Symplectic group.

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics
Finite groups Classification of finite simple groups cyclic alternating Lie type sporadic Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Symmetric group Sn Klein four-group V Dihedral group Dn Quaternion group Q Dicyclic group Dicn
Discrete groups Lattices Integers (Z) Free group Modular groups PSL(2,Z) SL(2,Z) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
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In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers. They are named after early 19th century mathematician Niels Henrik Abel.^[1]

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. 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.

Algebraic structures
Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Semiring Near-ring Commutative ring Integral domain Field Division ring Ring theory
Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map of lattices Lattice theory
Module-like Module Group with operators Vector space Linear algebra
Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra
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Contents

• 1 Definition
• 2 Facts
  • 2.1 Notation
  • 2.2 Multiplication table
• 3 Examples
• 4 Historical remarks
• 5 Properties
• 6 Finite abelian groups
  • 6.1 Classification
  • 6.2 Automorphisms
• 7 Finitely generated abelian groups
• 8 Infinite abelian groups
  • 8.1 Torsion groups
  • 8.2 Torsion-free and mixed groups
  • 8.3 Invariants and classification
  • 8.4 Additive groups of rings
• 9 Relation to other mathematical topics
• 10 A note on the typography
• 11 See also
• 12 Notes
• 13 References
• 14 External links

Definition[edit]

Group-like structures

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