# Abelian group
An abelian group is a group whose operation is commutative: a group \(G\) is abelian (or commutative) if \(xy = yx\) for all \(x, y \in G\)[1](https://crypto.stanford.edu/pbc/notes/group/group.html). The name honors the Norwegian mathematician [Niels Henrik Abel](https://www.edgepedia.org/niels-henrik-abel). Abelian groups are the simplest and most completely understood class of groups, and they serve as the basic objects of homological algebra, module theory, and algebraic topology.[2](https://arxiv.org/abs/2001.10469)
| Key fact | Detail |
|---|---|
| Defining property | \(xy = yx\) for all elements \(x, y\)[1](https://crypto.stanford.edu/pbc/notes/group/group.html) |
| Standard notation | Additive notation is often used: the identity is written \(0\) and the operation as \(a + b\)[3](https://crypto.stanford.edu/pbc/notes/group/abelian.html) |
| Structure theorem | Every finitely generated abelian group is a direct sum \(\mathbb{Z}^r \oplus \mathbb{Z}_{h_1} \oplus \cdots \oplus \mathbb{Z}_{h_n}\) with \(h_i \mid h_{i+1}\)[4](https://crypto.stanford.edu/pbc/notes/group/fgabelian.html) |
| Uniqueness | The invariant factors and elementary divisors of a finite abelian group are uniquely defined[5](https://people.math.osu.edu/leibman.1/algebra/groups.pdf) |
| Free groups | A free abelian group of rank \(n\) is one isomorphic to \(\mathbb{Z}^n\), and the rank is unique[6](https://www.math.uwaterloo.ca/~snew/pmath336-2024-S/Notes/Chap6FiniteAbelianGroups.pdf) |
## Definition and notation
A group is a set with an associative operation, an identity element, and inverses; the abelian condition adds commutativity. Because the operation commutes, additive notation is often used instead of multiplicative notation: the identity is represented by \(0\), and \(a + b\) represents the element obtained by applying the group operation to \(a\) and \(b\)[3](https://crypto.stanford.edu/pbc/notes/group/abelian.html). In this notation \(sg\) abbreviates the sum \(g + \cdots + g\) with \(s\) terms[7](https://cs.nyu.edu/spencer/algebra06/abelian.pdf).
Familiar examples include the integers \(\mathbb{Z}\) under addition, the integers modulo \(n\) under addition or multiplication (for suitable \(n\))[8](https://archive.nptel.ac.in/content/storage2/courses/106103015/module3/lec1/1.html), and the nonzero elements \(F^{\ast} = F \setminus \{0\}\) of a field \(F\), which form an abelian group under multiplication with identity \(1\) and inverse \(a^{-1}\) for each nonzero \(a\)[9](https://ntouikan.ext.unb.ca/MATH4043/groups-section-definitions.html). Not every group is abelian: matrix groups such as the general linear group provide standard noncommutative examples[10](https://www.youtube.com/watch?v=z9Ntoo-Ko18).
## Direct sums
A central construction for abelian groups is the direct sum. A group \(G\) is the direct sum of two subgroups \(U\) and \(V\), written \(G = U \oplus V\), if every element \(x \in G\) can be written \(x = u + v\) with \(u \in U\), \(v \in V\), and \(u + v = 0\) implies \(u = v = 0\)[3](https://crypto.stanford.edu/pbc/notes/group/abelian.html). The summands then intersect only in the identity, and the decomposition is unique: if \(u_1 + v_1 = u_2 + v_2\), then \(u_1 - u_2 = v_2 - v_1 \in U \cap V = \{0\}\)[3](https://crypto.stanford.edu/pbc/notes/group/abelian.html).
For a decomposition \(G = U_1 \oplus \cdots \oplus U_r\), every element of \(G\) has a unique expression \(u_1 + \cdots + u_r\), and if \(G\) is finite its order factors as \(|G| = |U_1| \cdots |U_r|\)[3](https://crypto.stanford.edu/pbc/notes/group/abelian.html). Direct sums are the vehicle in which the classification theorems below are stated.
## Free abelian groups
An abelian group \(A\) is a free abelian group of rank \(r\) if there exist \(u_1, \ldots, u_r \in A\) such that \(A = \langle u_1, \ldots, u_r \rangle\) and \(a_1 u_1 + \cdots + a_r u_r = 0\) implies \(a_1 = \cdots = a_r = 0\); equivalently, every element of \(A\) can be written uniquely as \(a_1 u_1 + \cdots + a_r u_r\)[3](https://crypto.stanford.edu/pbc/notes/group/abelian.html). The trivial group counts as the free abelian group of rank zero, generated by the empty set[3](https://crypto.stanford.edu/pbc/notes/group/abelian.html). In the other standard formulation, a free abelian group of rank \(n\) is an abelian group isomorphic to \(\mathbb{Z}^n\)[6](https://www.math.uwaterloo.ca/~snew/pmath336-2024-S/Notes/Chap6FiniteAbelianGroups.pdf).
The rank is an invariant of the group: if \(G \cong \mathbb{Z}^n\) and \(G \cong \mathbb{Z}^m\), then \(n = m\)[6](https://www.math.uwaterloo.ca/~snew/pmath336-2024-S/Notes/Chap6FiniteAbelianGroups.pdf). The standard proof reduces modulo 2: an isomorphism \(\mathbb{Z}^n \to \mathbb{Z}^m\) carries \(2\mathbb{Z}^n\) bijectively to \(2\mathbb{Z}^m\), inducing an isomorphism \(\mathbb{Z}^n/2\mathbb{Z}^n \cong \mathbb{Z}^m/2\mathbb{Z}^m\); since \(\mathbb{Z}^n/2\mathbb{Z}^n \cong \mathbb{Z}_2^n\), comparing orders gives \(2^n = 2^m\), so \(n = m\)[6](https://www.math.uwaterloo.ca/~snew/pmath336-2024-S/Notes/Chap6FiniteAbelianGroups.pdf).
## The structure theorem for finitely generated abelian groups
The classification of finitely generated abelian groups proceeds by presenting such a group as a quotient of a free group. If \(A = \langle a_1, \ldots, a_m \rangle\) is generated by \(m\) elements, the free abelian group \(F = \langle u_1, \ldots, u_m \rangle\) of rank \(m\) maps onto \(A\) by sending \(u_i\) to \(a_i\), and the first isomorphism theorem gives \(A \cong F/R\) for some subgroup \(R\) of \(F\)[4](https://crypto.stanford.edu/pbc/notes/group/fgabelian.html). A basis \(v_1, \ldots, v_m\) of \(F\) can be chosen so that \(R = \langle h_1 v_1, \ldots, h_q v_q \rangle\) with \(h_i \mid h_{i+1}\), \(h_i \ge 1\), and \(q \le m\)[4](https://crypto.stanford.edu/pbc/notes/group/fgabelian.html).
The resulting classification states that every finitely generated abelian group can be expressed as a direct sum of cyclic groups,
\[ A = \mathbb{Z}^r \oplus \mathbb{Z}_{h_1} \oplus \cdots \oplus \mathbb{Z}_{h_n} \]
where \(h_i \mid h_{i+1}\)[4](https://crypto.stanford.edu/pbc/notes/group/fgabelian.html). The number \(r\) is the rank of \(A\), the orders \(h_1, \ldots, h_n\) of the finite cyclic summands are its invariants, and \(A\) is finite if and only if its rank is zero[4](https://crypto.stanford.edu/pbc/notes/group/fgabelian.html). The theorem comes with a corollary for the torsion-free situation: being free is equivalent, for a finitely generated abelian group, to being torsion-free—meaning its only element of finite order is the identity[4](https://crypto.stanford.edu/pbc/notes/group/fgabelian.html).
The decomposition is unique. If a finitely generated abelian group has two decompositions \(A = \mathbb{Z}^r \oplus \mathbb{Z}_{e_1} \oplus \cdots \oplus \mathbb{Z}_{e_n} = \mathbb{Z}^s \oplus \mathbb{Z}_{d_1} \oplus \cdots \oplus \mathbb{Z}_{d_m}\), both with divisibility chains \(e_i \mid e_{i+1}\) and \(d_i \mid d_{i+1}\), then \(r = s\), \(n = m\), and \(e_i = d_i\) for all \(i\)[4](https://crypto.stanford.edu/pbc/notes/group/fgabelian.html). Correspondingly, for every finite abelian group its invariant factors and its elementary divisors are uniquely defined[5](https://people.math.osu.edu/leibman.1/algebra/groups.pdf).
## Finite abelian groups
For finite groups the theorem takes a particularly concrete form. Equivalently, every nontrivial finite abelian group \(A\) is isomorphic to a product of \(p\)-groups, one for each prime dividing its order[11](https://www.maths.tcd.ie/~donmoore/111notes.pdf).
The classification makes counting groups of a given order a finite arithmetic problem.
## Wider context
The structure theorem for finitely generated abelian groups is a specialization of the [structure theorem for finitely generated modules over a principal ideal domain](https://www.edgepedia.org/structure-theorem-for-finitely-generated-modules-over-a-principal-ideal-domain)[12](https://ebooks.lpude.in/arts/ma_mathematics/year_1/DMTH403_ABSTRACT_ALGEBRA.pdf). The theory extends beyond the finitely generated case: abelian groups are no longer assumed finite in general treatments, and the algebraic theory of abelian groups, including free groups and subgroups of free groups, underlies applications in algebraic topology[3](https://crypto.stanford.edu/pbc/notes/group/abelian.html) • [2](https://arxiv.org/abs/2001.10469). Subgroup structure is also well controlled: if \(F\) is a finitely generated free abelian group of rank \(r\), every subgroup of rank \(s\) with \(0 < s \le r\) is itself free[3](https://crypto.stanford.edu/pbc/notes/group/abelian.html).
## References
1. [Group Theory – Groups, Stanford (P.B. notes)](https://crypto.stanford.edu/pbc/notes/group/group.html)
2. [Algebraic theory of abelian groups, arXiv:2001.10469](https://arxiv.org/abs/2001.10469)
3. [Group Theory – Abelian Groups, Stanford (P.B. notes)](https://crypto.stanford.edu/pbc/notes/group/abelian.html)
4. [Group Theory – Finitely Generated Abelian Groups, Stanford (P.B. notes)](https://crypto.stanford.edu/pbc/notes/group/fgabelian.html)
5. [Leibman, Groups (Ohio State lecture notes)](https://people.math.osu.edu/leibman.1/algebra/groups.pdf)
6. [Chapter 6. The Classification of Finite Abelian Groups, PMATH 336, University of Waterloo](https://www.math.uwaterloo.ca/~snew/pmath336-2024-S/Notes/Chap6FiniteAbelianGroups.pdf)
7. [Classification of Finite Abelian Groups, NYU (Courant) class notes](https://cs.nyu.edu/spencer/algebra06/abelian.pdf)
8. [NPTEL course notes: Abelian groups](https://archive.nptel.ac.in/content/storage2/courses/106103015/module3/lec1/1.html)
9. [AATA Section 7.1: Definition and Basic Examples](https://ntouikan.ext.unb.ca/MATH4043/groups-section-definitions.html)
10. [(Abstract Algebra 1) Definition of an Abelian Group, learnifyable](https://www.youtube.com/watch?v=z9Ntoo-Ko18)
11. [111: Algebra notes, Trinity College Dublin](https://www.maths.tcd.ie/~donmoore/111notes.pdf)
12. [DMTH403 Abstract Algebra, Unit 10: Finite Abelian Groups, Lovely Professional University](https://ebooks.lpude.in/arts/ma_mathematics/year_1/DMTH403_ABSTRACT_ALGEBRA.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Group theory › Group structures and subgroups*
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