Direct product of groups

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In group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted GH. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.

Definition[edit]

Given groups G and H, the direct product G × H is defined as follows:

  1. The elements of G × H are ordered pairs (g, h), where gG and hH. That is, the set of elements of G × H is the Cartesian product of the sets G and H.
  2. The binary operation on G × H is defined componentwise:
    (g1, h1) · (g2, h2)  =  (g1 · g2, h1 · h2)

The resulting algebraic object satisfies the axioms for a group. Specifically:

Associativity
The binary operation on G × H is indeed associative.
Identity
The direct product has an identity element, namely (1G, 1H), where 1G is the identity element of G and 1H is the identity element of H.
Inverses
The inverse of an element (g, h) of G × H is the pair (g−1, h−1), where g−1 is the inverse of g in G, and h−1 is the inverse of h in H.

Examples[edit]

  • Let R be the group of real numbers under addition. Then the direct product R × R is the group of all two-component vectors (x, y) under the operation of vector addition:
    (x1, y1) + (x2, y2)  =  (x1 + x2, y1 + y2).
G
* 1 a
1 1 a
a a 1
H
* 1 b
1 1 b
b b 1
Then the direct product G × H is isomorphic to the Klein four-group:
G × H
* (1, 1) (a, 1) (1, b) (a, b)
(1, 1) (1, 1) (a, 1) (1, b) (a, b)
(a, 1) (a, 1) (1, 1) (a, b) (1, b)
(1, b) (1, b) (a, b) (1, 1) (a, 1)
(a, b) (a, b) (1, b) (a, 1) (1, 1)

Elementary properties[edit]

  • The order of a direct product G × H is the product of the orders of G and H:
| G × H |  =  | G | | H |.
This follows from the formula for the cardinality of the cartesian product of sets.
| (g, h) |  =  lcm( | g |, | h | ).
In particular, if | g | and | h | are relatively prime, then the order of (g, h) is the product of the orders of g and h .
  • As a consequence, if G and H are cyclic groups whose orders are relatively prime, then G × H is cyclic as well. That is, if m and n are relatively prime, then
( Z / mZ ) × ( Z / nZ )    Z / mnZ.
This fact is closely related to the Chinese remainder theorem.

Algebraic structure[edit]

Let G and H be groups, let P = G × H, and consider the following two subsets of P:

G′ = { (g, 1) : gG }    and    H′ = { (1, h) : hH }

Both of these are in fact subgroups of P, the first being isomorphic to G, and the second being isomorphic to H. If we identify these with G and H, respectively, then we can think of the direct product P as containing the original groups G and H as subgroups.

These subgroups of P have the following three important properties: (Saying again that we identify G and H with G and H, respectively.)

  1. The intersection GH is trivial.
  2. Every element of P can be expressed as the product of an element of G and an element of H.
  3. Every element of G commutes with every element of H.

Together, these three properties completely determine the algebraic structure of the direct product P. That is, if P is any group having subgroups G and H that satisfy the properties above, then P is necessarily isomorphic to the direct product of G and H. In this situation, P is sometimes referred to as the internal direct product of its subgroups G and H.

In some contexts, the third property above is replaced by the following:

3′.  Both G and H are normal in P.

This property is equivalent to property 3, since the elements of two normal subgroups with trivial intersection necessarily commute, a fact which can be deduced by considering the commutator [g,h] of any g in G, h in H.

Examples[edit]

V
* 1 a b c
1 1 a b c
a a 1 c b
b b c 1 a
c c b a 1
Then V is the internal direct product of the two-element subgroups { 1, a } and { 1, b }.
  • Let a be a cyclic group of order mn, where m and n are relatively prime. Then an and am are cyclic subgroups of orders m and n, respectively, and a is the internal direct product of these subgroups.
  • Similarly, when n is odd the orthogonal group O(n, R) is the internal direct product of the special orthogonal group SO(n, R) and the two-element subgroup { −I, I }, where I denotes the identity matrix.
  • The symmetry group of a cube is the internal direct product of the subgroup of rotations and the two-element group { −I, I }, where I is the identity element and I is the point reflection through the center of the cube. A similar fact holds true for the symmetry group of an icosahedron.
D4n = 〈 r, s | r2n = s2 = 1, sr = r−1s.
Then D4n is the internal direct product of the subgroup r2, s (which is isomorphic to D2n) and the two-element subgroup { 1, rn }.

Presentations[edit]

The algebraic structure of G × H can be used to give a presentation for the direct product in terms of the presentations of G and H. Specifically, suppose that

G = 〈 SG | RG     and     H = 〈 SH | RH,

where SG and SH are (disjoint) generating sets and RG and RH are defining relations. Then

G × H = 〈 SGSH | RGRHRP

where RP is a set of relations specifying that each element of SG commutes with each element of SH

For example, suppose that

G = 〈 a | a3 = 1 〉     and     H = 〈 b | b5 = 1 〉.

Then

G × H = 〈 a, b | a3 = 1, b5 = 1, ab = ba.

Normal structure[edit]

As mentioned above, the subgroups G and H are normal in G × H. Specifically, define functions πG: G × HG and πH: G × HH by

πG(g, h) = g     and     πH(g, h) = h.

Then πG and πH are homomorphisms, known as projection homomorphisms, whose kernels are H and G, respectively.

It follows that G × H is an extension of G by H (or vice-versa). In the case where G × H is a finite group, it follows that the composition factors of G × H are precisely the union of the composition factors of G and the composition factors of H.

Further properties[edit]

Universal property[edit]

The direct product G × H can be characterized by the following universal property. Let πG: G × HG and πH: G × HH be the projection homomorphisms. Then for any group P and any homomorphisms ƒG: PG and ƒH: PH, there exists a unique homomorphism ƒ: PG × H making the following diagram commute:

DirectProductDiagram.png

Specifically, the homomorphism ƒ is given by the formula

ƒ(p)  =  ( ƒG(p), ƒH(p) ).

This is a special case of the universal property for products in category theory.

Subgroups[edit]

If A is a subgroup of G and B is a subgroup of H, then the direct product A × B is a subgroup of G × H. For example, the isomorphic copy of G in G × H is the product G × {1}, where {1} is the trivial subgroup of H.

If A and B are normal, then A × B is a normal subgroup of G × H. Moreover, the quotient (G × H) / (A × B) is isomorphic to the direct product of the quotients G / A and H / B:

(G × H) / (A × B)    (G / A) × (H / B).

Note that it is not true in general that every subgroup of G × H is the product of a subgroup of G with a subgroup of H. For example, if G is any group, then the product G × G has a diagonal subgroup

Δ  =  { (g, g) : gG }

which is not the direct product of two subgroups of G. Other subgroups include fiber products of G and H (see below). The subgroups of direct products are described by Goursat's lemma.

Conjugacy and centralizers[edit]

Two elements (g1, h1) and (g2, h2) are conjugate in G × H if and only if g1 and g2 are conjugate in G and h1 and h2 are conjugate in H. It follows that each conjugacy class in G × H is simply the Cartesian product of a conjugacy class in G and a conjugacy class in H.

Along the same lines, if (g, h) ∈ G × H, the centralizer of (g, h) is simply the product of the centralizers of g and h:

CG×H(g, h)  =  CG(g) × CH(h).

Similarly, the center of G × H is the product of the centers of G and :

Z(G × H)  =  Z(G) × Z(H).

Normalizers behave in a more complex manner since not all subgroups of direct products themselves decompose as direct products.

Automorphisms and endomorphisms[edit]

If α is an automorphism of G and β is an automorphism of H, then the product function α × β: G × HG × H defined by

(α × β)(g, h) = (α(g), β(h))

is an automorphism of G × H. It follows that Aut(G × H) has a subgroup isomorphic to the direct product Aut(G) × Aut(H).

It is not true in general that every automorphism of G × H has the above form. (That is, Aut(G) × Aut(H) is often a proper subgroup of Aut(G × H).) For example, if G is any group, then there exists an automorphism σ of G × G that switches the two factors, i.e.

σ(g1, g2) = (g2, g1).

For another example, the automorphism group of Z × Z is GL(2, Z), the group of all 2 × 2 matrices with integer entries and determinant ±1. This automorphism group is infinite, but only finitely many of the automorphisms have the form given above.

In general, every endomorphism of G × H can be written as a 2 × 2 matrix

\begin{bmatrix}\alpha & \beta \\ \gamma & \delta\end{bmatrix}

where α is an endomorphism of G, δ is an endomorphism of H, and β: HG and γ: GH are homomorphisms. Such a matrix must have the property that every element in the image of α commutes with every element in the image of β, and every element in the image of γ commutes with every element in the image of δ.

When G and H are indecomposable, centerless groups, then the automorphism group is relatively straightforward, being Aut(G) × Aut(H) if G and H are not isomorphic, and Aut(G) wr 2 if GH, wr denotes the wreath product. This is part of the Krull–Schmidt theorem, and holds more generally for finite direct products.

Generalizations[edit]

Finite direct products[edit]

It is possible to take the direct product of more than two groups at once. Given a finite sequence G1, ..., Gn of groups, the direct product

\prod_{i=1}^n G_i \;=\; G_1 \times G_2 \times \cdots \times G_n

is defined as follows:

  • The elements of G1 × ··· × Gn are tuples (g1, ..., gn), where giGi for each i.
  • The operation on G1 × ··· × Gn is defined componentwise:
    (g1, ..., gn)(g1′, ..., gn′)  =  (g1g1′, ..., gngn′).

This has many of the same properties as the direct product of two groups, and can be characterized algebraically in a similar way.

Infinite direct products[edit]

It is also possible to take the direct product of an infinite number of groups. For an infinite sequence G1, G2, ... of groups, this can be defined just like the finite direct product of above, with elements of the infinite direct product being infinite tuples.

More generally, given an indexed familyGi }iI of groups, the direct product iI Gi is defined as follows:

  • The elements of iI Gi are the elements of the infinite Cartesian product of the sets Gi, i.e. functions ƒ: I UiI Gi with the property that ƒ(i) ∈ Gi for each i.
  • The product of two elements ƒ, g is defined componentwise:
    (ƒ • g)(i)  =  ƒ(i) • g(i).

Unlike a finite direct product, the infinite direct product iI Gi is not generated by the elements of the isomorphic subgroups { Gi }iI. Instead, these subgroups generate a subgroup of the direct product known as the infinite direct sum, which consists of all elements that have only finitely many non-identity components.

Other products[edit]

Semidirect products[edit]

Main article: Semidirect product

Recall that a group P with subgroups G and H is isomorphic to the direct product of G and H as long as it satisfies the following three conditions:

  1. The intersection GH is trivial.
  2. Every element of P can be expressed as the product of an element of G and an element of H.
  3. Both G and H are normal in P.

A semidirect product of G and H is obtained by relaxing the third condition, so that only one of the two subgroups G, H is required to be normal. The resulting product still consists of ordered pairs (g, h), but with a slightly more complicated rule for multiplication.

It is also possible to relax the third condition entirely, requiring neither of the two subgroups to be normal. In this case, the group P is referred to as a Zappa–Szép product of G and H.

Free products[edit]

Main article: Free product

The free product of G and H, usually denoted GH, is similar to the direct product, except that the subgroups G and H of GH are not required to commute. That is, if

G = SG | RG     and     H = SH | RH,

are presentations for G and H, then

GH = SGSH | RGRH.

Unlike the direct product, elements of the free product cannot be represented by ordered pairs. In fact, the free product of any two nontrivial groups is infinite. The free product is actually the coproduct in the category of groups.

Subdirect products[edit]

Main article: Subdirect product

If G and H are groups, a subdirect product of G and H is any subgroup of G × H which maps surjectively onto G and H under the projection homomorphisms. By Goursat's lemma, every subdirect product is a fiber product, and vice versa.

Fiber products[edit]

Let G, H, and Q be groups, and let φ: GQ and χ: HQ be epimorphisms. The fiber product of G and H over Q, also known as a pullback, is the following subgroup of G × H:

G ×Q H  =  { (g, h) ∈ G × H : φ(g) = χ(h) }.

By Goursat's lemma, every subdirect product is a fiber product, and vice versa.

References[edit]

  1. ^ Gallian, Joseph A. (2010). Contemporary Abstract Algebra (7 ed.). Cengage Learning. p. 157. ISBN 9780547165097.