Direct product of groups

In mathematics, specifically 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 ${\displaystyle G\oplus H}$. 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

Given groups G (with operation *) and H (with operation ∆), the direct product G × H is defined as follows:

1. The underlying set is the Cartesian product, G × H. That is, the ordered pairs (g, h), where g ∈ G and h ∈ H.
2. The binary operation on G × H is defined component-wise:

   (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 associative.

Identity

The direct product has an identity element, namely (1_G, 1_H), where 1_G is the identity element of G and 1_H 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

• 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).

• Let R⁺ be the group of positive real numbers under multiplication. Then the direct product R⁺ × R⁺ is the group of all vectors in the first quadrant under the operation of component-wise multiplication

(x₁, y₁) × (x₂, y₂)  =  (x₁ × x₂,  y₁ × y₂).

• Let G and H be cyclic groups with two elements each:

• ${\displaystyle G}$ 

  *	1	a
  1	1	a
  a	a	1

• ${\displaystyle H}$ 

  *	1	b
  1	1	b
  b	b	1

Then the direct product G × H is isomorphic to the Klein four-group:

${\displaystyle G\times H}$ 

*	(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

• The direct product is commutative and associative up to isomorphism. That is, G × H ≅ H × G and (G × H) × K ≅ G × (H × K) for any groups G, H, and K.
• The trivial group is the identity element of the direct product, up to isomorphism. If E denotes the trivial group, G ≅ G × E ≅ E × G for any groups G.
• 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.
• The order of each element (g, h) is the least common multiple of the orders of g and h:^[1]

  |(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

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

G′ = { (g, 1) : g ∈ G }    and    H′ = { (1, h) : h ∈ H }.

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 G ∩ H is trivial.
2. Every element of P can be expressed uniquely 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

• Let V be the Klein four-group:

  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 ${\displaystyle \langle a\rangle }$  be a cyclic group of order mn, where m and n are relatively prime. Then ${\displaystyle \langle a^{n}\rangle }$  and ${\displaystyle \langle a^{m}\rangle }$  are cyclic subgroups of orders m and n, respectively, and ${\displaystyle \langle a\rangle }$  is the internal direct product of these subgroups.
• Let C^× be the group of nonzero complex numbers under multiplication. Then C^× is the internal direct product of the circle group T of unit complex numbers and the group R⁺ of positive real numbers under multiplication.
• If n is odd, then the general linear group GL(n, R) is the internal direct product of the special linear group SL(n, R) and the subgroup consisting of all scalar matrices.
• 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.
• Let n be odd, and let D_4n be the dihedral group of order 4n:

  ${\displaystyle D_{4n}=\langle r,s\mid r^{2n}=s^{2}=1,sr=r^{-1}s\rangle .}$ 

  Then D_4n is the internal direct product of the subgroup ${\displaystyle \langle r^{2},s\rangle }$  (which is isomorphic to D_2n) and the two-element subgroup {1, rⁿ}.

Presentations

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

${\displaystyle G=\langle S_{G}\mid R_{G}\rangle \ \ }$  and ${\displaystyle \ \ H=\langle S_{H}\mid R_{H}\rangle ,}$ 

where ${\displaystyle S_{G}}$  and ${\displaystyle S_{H}}$  are (disjoint) generating sets and ${\displaystyle R_{G}}$  and ${\displaystyle R_{H}}$  are defining relations. Then

${\displaystyle G\times H=\langle S_{G}\cup S_{H}\mid R_{G}\cup R_{H}\cup R_{P}\rangle }$ 

where ${\displaystyle R_{P}}$  is a set of relations specifying that each element of ${\displaystyle S_{G}}$  commutes with each element of ${\displaystyle S_{H}}$ .

For example if

${\displaystyle G=\langle a\mid a^{3}=1\rangle \ \ }$  and ${\displaystyle \ \ H=\langle b\mid b^{5}=1\rangle }$ 

then

${\displaystyle G\times H=\langle a,b\mid a^{3}=1,b^{5}=1,ab=ba\rangle .}$ 

Normal structure

As mentioned above, the subgroups G and H are normal in G × H. Specifically, define functions π_G: G × H → G and π_H: G × H → H by

π_G(g, h) = g     and     π_H(g, h) = h.

Then π_G and π_H 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

Universal property

Main article: Product (category theory)

The direct product G × H can be characterized by the following universal property. Let π_G: G × H → G and π_H: G × H → H be the projection homomorphisms. Then for any group P and any homomorphisms ƒ_G: P → G and ƒ_H: P → H, there exists a unique homomorphism ƒ: P → G × H making the following diagram commute:

 

Specifically, the homomorphism ƒ is given by the formula

ƒ(p)  =  ( ƒ_G(p), ƒ_H(p) ).

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

Subgroups

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 of the direct products is isomorphic to the direct product of the quotients:

(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 non-trivial group, then the product G × G has a diagonal subgroup

Δ = { (g, g) : g ∈ G }

which is not the direct product of two subgroups of G.

The subgroups of direct products are described by Goursat's lemma. Other subgroups include fiber products of G and H.

Conjugacy and centralizers

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:

C_G×H(g, h)  =  C_G(g) × C_H(h).

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

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

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

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

where α is an endomorphism of G, δ is an endomorphism of H, and β: H → G and γ: G → H 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 G ≅ H, wr denotes the wreath product. This is part of the Krull–Schmidt theorem, and holds more generally for finite direct products.

Generalizations

Finite direct products

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

${\displaystyle \prod _{i=1}^{n}G_{i}\;=\;G_{1}\times G_{2}\times \cdots \times G_{n}}$ 

is defined as follows:

• The elements of G₁ × ⋯ × G_n are tuples (g₁, ..., g_n), where g_i ∈ G_i for each i.
• The operation on G₁ × ⋯ × G_n is defined component-wise:

  (g₁, ..., g_n)(g₁′, ..., g_n′) = (g₁g₁′, ..., g_ng_n′).

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

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 family { G_i }_i∈I of groups, the direct product Π_i∈I G_i is defined as follows:

• The elements of Π_i∈I G_i are the elements of the infinite Cartesian product of the sets G_i; i.e., functions ƒ: I → ⋃_i∈I G_i with the property that ƒ(i) ∈ G_i 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 Π_i∈I G_i is not generated by the elements of the isomorphic subgroups { G_i }_i∈I. 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

Semidirect products

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 G ∩ H is trivial.
2. Every element of P can be expressed uniquely 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

Main article: Free product

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

G = 〈 S_G | R_G 〉     and     H = 〈 S_H | R_H 〉,

are presentations for G and H, then

G ∗ H = 〈 S_G ∪ S_H | R_G ∪ R_H 〉.

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

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.

Fiber products

Main article: Pullback (category theory)

Let G, H, and Q be groups, and let φ: G → Q and χ: H → Q be homomorphisms. 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) }.

If φ: G → Q and χ: H → Q are epimorphisms, then this is a subdirect product.

References

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