Abelian group

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In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that

a * b = b * a

for all a and b in G. In other words, the order of elements in a product doesn't matter. Such groups are generally easier to understand.

Abelian groups are named after Niels Henrik Abel. Groups that are not commutative are called non-abelian (rather than non-commutative).

Notation

There are two main notational conventions for abelian groups -- additive and multiplicative.

Convention Operation Identity Powers Inverse Direct sum/product
Addition a + b 0 na a GH
Multiplication a * b or ab e or 1 an a−1 G × H

The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. When studying abelian groups in their own right, the additive notation is usually used.

Examples

Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. In particular, the integers Z form an abelian group under addition, as do the integers modulo n Z/nZ.

The real numbers form an abelian group under addition, as do the non-zero real numbers under multiplication.

Every ring is an abelian group with respect to its addition operator. Also, in every commutative ring the invertible elements, or units form an abelian multiplicative group.

Any subgroup of an abelian group is normal, so for every subgroup there is a quotient group. Subgroups, factor groups, and direct sums of abelian groups are again abelian.

Multiplication table

To verify that a certain finite group is indeed abelian, a table (matrix) can be drawn up in the similar fashion to a multiplication table, where, if the group is G = {g1 = e, g2, ..., gn} under the operation ⋅, the (i, j)'th entry of this table contains the product gigj. The group is abelian if and only if this table is symmetric about the main diagonal (i.e. if the matrix is a symmetric matrix).

This is true since if the group is abelian, then gigj = gjgi. This implies that the (i, j)'th entry of the table equals the (j, i)'th entry - i.e. the table is symmetric about the main diagonal.

Properties

If n is a natural number and x is an element of an abelian group G written additively, then nx can be defined as x + x + ... + x (n summands) and (−n)x = −(nx). 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.e. modules over the principal ideal domain Z) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups.

If f, g : G  →  H are two group homomorphisms between abelian groups, then their sum f + g, defined by (f + g)(x) = f(x) + g(x), is again a homomorphism. (This is not true if H is a non-abelian group). The set Hom(G, H) 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 spaces, every abelian group has a rank. It is defined as the cardinality of the largest set of linearly independent elements of the group. The integers and the rational numbers have rank one, as well as every subgroup of the rationals. While the rank one torsion-free abelian groups are well understood, even finite-rank abelian groups are not well understood. Infinite-rank abelian groups can be extremely complex and many open questions exist, often intimately connected to questions of set theory.

Finite abelian groups

The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order. This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0.

Zmn is isomorphic to the direct product of Zm and Zn if and only if m and n are coprime.

Therefore we can write any abelian group G as a direct product of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_{k_1} \oplus \cdots \oplus \mathbb{Z}_{k_u}}

in two unique ways:

  • where the numbers k1,...,ku are powers of primes
  • where k1 divides k2, which divides k3 and so on up to ku.

Thus we have 3 2 or 6, 5 2 or 10, 4 3 or 12, 3 2 2 or 6 2, 7 2 or 14, and 5 3 or 15, but anyway 2 2, 4 2, 2 2 2, 3 3, 8 2, 4 4, 4 2 2, and 2 2 2 2.

For example, Z/15Z = Z/15 can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: Z/15 = {0, 5, 10} ⊕ {0, 3, 6, 9, 12}. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic.

For another example, every group of order 8 is isomorphic to either Z/8 (the integers 0 to 7 under addition modulo 8), Z/4  ⊕ Z/2 (the odd integers 1 to 15 under multiplication modulo 16), or Z/2  ⊕  Z/2  ⊕  Z/2.

List of small abelian groups

Extracted from the list of small groups is the following table of small abelian groups.

Note that e.g. "3 × Z2" means that there are 3 subgroups of type Z2, while elsewhere the cross means direct product.

Template:Groups 16a

Order Group Subgroups Properties Cycle graph
1 trivial group = Z1 = S1 = A2 - various properties hold trivially
2 Z2 = S2 = Dih1 - simple, the smallest non-trivial group
3 Z3 = A3 - simple
4 Z4 Z2   
Klein four-group = Z2 2 = Dih2 3 × Z2 the smallest non-cyclic group
5 Z5 - simple
6 Z6 = Z3 × Z2 Z3 , Z2  
7 Z7 - simple
8 Z8 Z4 , Z2  
Z4 ×Z2 2 × Z4 , Z22, 3 ×Z2  
Z2 3 7 × Z22 , 7 × Z2 the non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines
9 Z9 Z3  
Z3 × Z3 4 × Z3  
10 Z10 = Z5 × Z2 Z5 , Z2  
11 Z11 - simple
12 Z12 = Z4 × Z3 Z6 , Z4 , Z3 , Z2  
Z6 × Z2 = Z3 × Z2 × Z2 = Z3 × Z22 2 × Z6, Z3 , 3 × Z2  
13 Z13 - simple
14 Z14 = Z7 × Z2 Z7 , Z2  
15 Z15 = Z5 × Z3 Z5 , Z3  
16

Relation to other mathematical topics

The abelian group, together with group homomorphisms, form a category, the prototype of an abelian category. In this encyclopedia, we denote this category Ab. See category of abelian groups for a list of its properties.

Many large abelian groups carry a natural topology, turning them into topological groups.

A note on the typography

Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in being expressed with a lowercase a, rather than A (compare, for example, Riemannian). Contrary to what one might expect, naming a concept in this way is considered one of the highest honors in mathematics for the namesake.

cs:Abelova grupa de:Abelsche Gruppe et:Abeli rühm es:Grupo abeliano eo:Komuta grupo fr:Groupe abélien ko:아벨군 it:Gruppo abeliano nl:Abelse groep ja:アーベル群 no:Abelsk gruppe pl:Grupa abelowa pt:Grupo abeliano ro:Grup abelian ru:Абелева группа sk:Abelovská grupa sl:Abelova grupa sv:Abelsk grupp zh:阿贝尔群