Abelian group

In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that the group operation * is commutative, so that for all a and b in G, a * b = b * a. Abelian groups are named after Norwegian mathematician Niels Henrik Abel. Groups in which the group operation is not commutative are called non-abelian (or non-commutative). Since the group operation in an abelian group is commutative as well as associative, the value of a product of group elements is independent of the order in which the product is calculated. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, although infinite abelian groups are the subject of current research.

Notation

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

The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. When studying abelian groups apart from other groups, the additive notation is usually used.

Examples

Every cyclic group G is abelian, because if x, y are in G, then xy = a^maⁿ = a^{m + n} = a^{n + m} = aⁿa^m = yx. Thus the integers, Z, form an abelian group under addition, as do the integers modulo n, Z/nZ.

Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.

Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian.

Matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative.

Multiplication table

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. If the group is G = {g₁ = e, g₂, ..., g_n} under the operation ⋅, the (i, j)'th entry of this table contains the product g_i ⋅ g_j. 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 g_i ⋅ g_j = g_j ⋅ g_i. 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.

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.

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_{mn}

is isomorphic to the direct product of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_m
and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_n
if and only if m and n are coprime.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_{k_1} \oplus \cdots \oplus \mathbb{Z}_{k_u}

in two unique ways:

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

For example, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}/15\mathbb{Z}\cong\mathbb{Z}_{15}

can be expressed as the direct sum of two cyclic subgroups of order 3 and 5:  Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_{15}\cong\{0, 5, 10\}\oplus\{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 abelian group of order 8 is isomorphic to either Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_8

(the integers 0 to 7 under addition modulo 8), Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_4\oplus\mathbb{Z}_2
(the odd integers 1 to 15 under multiplication modulo 16), or Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_2\oplus\mathbb{Z}_2\oplus\mathbb{Z}_2

.

See also list of small groups for finite abelian groups of order 16 or less.

Automorphisms of finite abelian groups

One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group G. To do this, one uses the fact (which will not be proved here) that if G splits as a direct sum H Failed to parse (Missing texvc executable; please see math/README to configure.): \oplus

K of subgroups of coprime order, then Aut(H Failed to parse (Missing texvc executable; please see math/README to configure.): \oplus
K) Failed to parse (Missing texvc executable; please see math/README to configure.): \cong
Aut(H) Failed to parse (Missing texvc executable; please see math/README to configure.): \oplus
Aut(K).

Given this, the fundamental theorem shows that to compute the automorphism group of G it suffices to compute the automorphism groups of the Sylow p-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of p). Fix a prime p and suppose the exponents e_i of the cyclic factors of the Sylow p-subgroup are arranged in increasing order:

Failed to parse (Missing texvc executable; please see math/README to configure.): e_1\leq e_2 \leq\cdots\leq e_n

for some n > 0. One needs to find the automorphisms of

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_{p^{e_1}} \oplus \cdots \oplus \mathbb{Z}_{p^{e_n}}

One special case is when n = 1, so that there is only one cyclic prime-power factor in the Sylow p-subgroup P. In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when n is arbitrary but e_i = 1 for 1 ≤ i ≤ n. Here, one is considering P to be of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p

,

so elements of this subgroup can be viewed as comprising a vector space of dimension n over the finite field of p elements Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{F}_p . The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{Aut}(P)\cong\mathrm{GL}(n,\mathbb{F}_p)

,

which is easily shown to have order

Failed to parse (Missing texvc executable; please see math/README to configure.): |\mathrm{Aut}(P)|=(p^n-1)\cdots(p^n-p^{n-1})

.

In the most general case, where the e_i and n are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines

Failed to parse (Missing texvc executable; please see math/README to configure.): d_k=\mathrm{max}\{r|e_r = e_k^{\,}\}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): c_k=\mathrm{min}\{r|e_r=e_k^{\,}\}

then one has in particular d_k ≥ k, c_k ≤ k, and

Failed to parse (Missing texvc executable; please see math/README to configure.): |\mathrm{Aut}(P)| = \left(\prod_{k=1}^n{p^{d_k} - p^{k-1}}\right)\left(\prod_{j=1}^n{(p^{e_j})^{n-d_j}}\right)\left(\prod_{i=1}^n{(p^{e_i-1})^{n-c_i+1}}\right).

One can check that this yields the orders in the previous examples as special cases (see [Hillar,Rhea]).

Relation to other mathematical topics

Many large abelian groups possess a natural topology, which turns them into topological groups.

The collection of all abelian groups, together with the homomorphisms between them, forms the category Ab, the prototype of an abelian category.

Nearly all well-known algebraic structures other than Boolean algebra, are undecidable. Hence it is surprising that Tarski's student Szmielew (1955) proved that the first order theory of abelian groups, unlike its nonabelian counterpart, is decidable. This decidability, plus the fundamental theorem of finite abelian groups described above, highlight some of the successes in abelian group theory, but there are still many areas of current research:

• Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood;
• There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
• While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the cased of countable mixed groups is much less mature.
• Many mild extensions of the first order theory of abelian groups are known to be undecidable.
• Finite abelian groups remain a topic of research in computational group theory.

Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is:

• Undecidable in ZFC, the conventional axiomatic set theory from which nearly all of present day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC;
• Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom;
• Decidable if ZFC is augmented with the axiom of constructibility (see statements true in L).

A note on the typography

Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is spelled with a lowercase a, rather than an uppercase A (cf. Riemannian). Contrary to what one might expect, naming a concept in this way is considered one of the highest honours in mathematics for the namesake.

See also

• Abelianization
• Class field theory
• Commutator subgroup
• Elementary abelian group
• Finitely generated abelian group
• Free abelian group
• Pontryagin duality
• Torsion-free abelian groups of rank 1

References

• Fuchs, László (1970) Infinite abelian groups, Vol. I. Pure and Applied Mathematics, Vol. 36. New York-London: Academic Press. xi+290 pp. MR0255673
• ------ (1973) Infinite abelian groups, Vol. II. Pure and Applied Mathematics. Vol. 36-II. New York-London: Academic Press. ix+363 pp. MR0349869
• Hillar, Christopher. and Rhea, Darren Automorphisms of Finite Abelian Groups.
• Szmielew, Wanda (1955) "Elementary properties of abelian groups," Fundamenta Mathematica 41: 203-71.bn:আবেলীয় গ্রুপ

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