Abelian group

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In abstract algebra, an abelian group is a group (G, *) that is commutative, i.e., in which a * b = b * a holds for all elements a and b in G. Abelian groups are named after Niels Henrik Abel.

If a group is abelian, we usually write the operation as + instead of *, the identity element as 0 (often called the zero element in this context) and the inverse of the element a as -a.

Examples of abelian groups include all cyclic groups such as the integers Z (with addition) and the integers modulo n Z_n (also with addition). The real numbers form an abelian group with addition, as do the non-zero real numbers with multiplication. Every field gives rise to two abelian groups in the same fashion. Another important example is the factor group Q/Z, an injective cogenerator.

To verify that a certain finite group is indeed abelian, a table can be drawn up in the similar fashion to a multiplication table, where, if the group is G={g₀=e, g₁,...,g_n} under the operation ⋅, the (i, j)'th row of this table contains the product g_i ⋅ g_j. If this table is symmetric about the main diagonal, then the group is abelian, and vice versa. This is true since if the group is abelian, then g_i ⋅ g_j = g_j ⋅ g_i. This implies that each entry in the (i, j)'th row of the table is the same as the entry in the (j, i)'th row of the table - meaning the table is symmetric about the main diagonal.

Table of contents

1 Properties
2 Finite abelian groups
3 Relation to other mathematical topics

Properties

If n is a natural number and x is an element of an abelian group G, 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 can often be generalized to theorems about modules over principal ideal domains. An example is the classification of finitely generated abelian groups.

Any subgroup of an abelian group is normal, and hence factor groups can be formed freely. Subgroups, factor groups, products and direct sums of abelian groups are again abelian. 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.

For example, Z₁₅ can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: Z₁₅ = {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.

To take another example, every group of order 8 is isomorphic to either Z₈ (the integers 0 to 7 under addition modulo 8), Z₄ × Z₂ (the odd integers 1 to 15 under multiplication modulo 16), or Z₂ × Z₂ × Z₂ (the power set of {1, 2, 3} under the symmetric difference operation).

The same concept can be extended to some infinite abelian groups. See finitely generated abelian group.

Relation to other mathematical topics

The abelian groups, together with group homomorphisms, form a category, the prototype of an abelian category. In Wikipedia, we denote this category with 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.