Abelian group

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, although large infinite abelian groups remain a subject of current research.

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

Notation

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

Convention	Operation	Identity	Powers	Inverse	Direct sum/product
Addition	x + y	0	nx	−x	G ⊕ H
Multiplication	x * y or xy	e or 1	xⁿ	x⁻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 = a^maⁿ = a^{m + n} = a^{n + m} = aⁿa^m = 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.

The matrices, in contrast, do not form an abelian group under multiplication, for even when restricted to sets of invertible matrices, matrix multplication is generally non-communtative.

Multiplication table

To verify that a certain finite group is indeed abelian, a table (matrix) - frequently known as a Cayley table - can be drawn up in the similar fashion to a multiplication table, where, 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. 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.

Z_mn is isomorphic to the direct product of Z_m and 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,...,k_u are powers of primes
where k₁ divides k₂, which divides k₃ and so on up to k_u.

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.

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 ⊕ K of subgroups of coprime order, then Aut(H ⊕ K) ≅ Aut(H) ⊕ 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 cyclic subgroups with order a power of p). Fix a prime p and suppose the exponents e_i of the cyclic factors are in arranged in increasing order:

e₁ ≤ e₂ ≤ … ≤ e_n

for some n > 0.

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

Z_p ⊕ … ⊕ 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 F_p. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

Aut(P) ≅ GL(n, F_p),

which is easily shown to have order

|Aut(P)| = (pⁿ − 1)…(pⁿ − pⁿ⁻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

d_k = max{r | e_r = e_k}

and

c_k = min{r | e_r = e_k}

then one has in particular d_k ≥ k, c_k ≤ 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 ^*).

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.

Order	Group	Subgroups	Properties
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
4	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
9	Z₃ × Z₃	4 × Z₃
10	Z₁₀ = Z₅ × Z₂	Z₅ , Z₂
11	Z₁₁	-	simple
12	Z₁₂ = Z₄ × Z₃	Z₆ , Z₄ , Z₃ , Z₂
12	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. 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.

Reference

Christopher Hillar and Darren Rhea. Automorphisms of Finite Abelian Groups.

Abelian group theory | Properties of groups

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