Abelian group

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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.

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[edit] Notation

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

Convention Operation Identity Powers Inverse
Addition x + y 0 nx −x
Multiplication x * y or xy e or 1 xn x −1

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.

[edit] Examples

Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = 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.

[edit] 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, ..., gn} under the operation ⋅, the (i, j)'th entry of this table contains the product gi ⋅ gj. 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 gi ⋅ gj = gj ⋅ gi. 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.

[edit] 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.

[edit] 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.

\mathbb{Z}_{mn} is isomorphic to the direct product of \mathbb{Z}_m and \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

\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.

For example, \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: \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 \mathbb{Z}_8 (the integers 0 to 7 under addition modulo 8), \mathbb{Z}_4\oplus\mathbb{Z}_2 (the odd integers 1 to 15 under multiplication modulo 16), or \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.

[edit] 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 \oplus K of subgroups of coprime order, then Aut(H \oplus K) \cong Aut(H) \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 ei of the cyclic factors of the Sylow p-subgroup are arranged in increasing order:

e_1\leq e_2 \leq\cdots\leq e_n

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

\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 ei = 1 for 1 ≤ i ≤ n. Here, one is considering P to be of the form

\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 \mathbb{F}_p. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

\mathrm{Aut}(P)\cong\mathrm{GL}(n,\mathbb{F}_p),

which is easily shown to have order

|\mathrm{Aut}(P)|=(p^n-1)\cdots(p^n-p^{n-1}).

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

d_k=\mathrm{max}\{r|e_r = e_k^{\,}\}

and

c_k=\mathrm{min}\{r|e_r=e_k^{\,}\}

then one has in particular dk ≥ k, ck ≤ k, and

|\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]).

[edit] 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:

[edit] 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, indicating how ubiquitous the concept is in modern mathematics.[1]

[edit] See also

[edit] References

  1. ^ Abel Prize Awarded: The Mathematicians' Nobel
  • 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. Amer. Math. Monthly 114 (2007), no. 10, 917-923. [1].
  • Szmielew, Wanda (1955) "Elementary properties of abelian groups," Fundamenta Mathematica 41: 203-71.