Direct product groups

In summary, the author is discussing how to represent elements of a group A as ordered pairs (a,b) and how to represent elements of group B as (a,e_B) where e_B is the element of B that is equal to the element a of A.
  • #1
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From Gauge Theory of particle physics, Cheng and Li I don't understand the flollowing:

"Given any two groups G={g1,..} H= {h1,h2,...}
if the g's commute with the h's we can define a direct product group G x H={[tex]g_ih_j[/tex]} with the multplication law:
[tex]g_kh_l . g_mh_n = g_kh_m . h_lh_n[/tex]

Examples of direct product groups are SU(2) x U(1) ( the group consists of elements which are direct products of SU(2) matrices and the U(1) phase factor) and
SU(3) x SU(3) (the group consists of elements which are direct products of matrices of two different SU(3)'s."

My question is: You need the g's and h's to commute. I can see how SU(2) and U(1) elements can commute. I don't see how an element of SU(3) commutes with
another SU(3) element?
 
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  • #2
foobar said:
"Given any two groups G={g1,..} H= {h1,h2,...}
if the g's commute with the h's
What he means is if they commute in the group we're about to form.

This statement, as written, doesn't make sense: if all we have is a group G and a group H, then we don't have a way to multiply an element of G with an element of H. But in the process of building a new group, we have lots of freedom to choose how such a multiplication might make sense.


I don't see how an element of SU(3) commutes with
another SU(3) element?
In particular, the issue here is that the elements of the different 'copies' of SU(3) commute, simply by the definition of the product group. If I paint one copy red and another copy blue, then we can have both
[tex]{\color{red} x y} \neq {\color{red} y x}[/tex]
and
[tex]{\color{red} x} {\color{blue} y} = {\color{blue} y} {\color{red} x} [/tex]
because the product is defined differently for elements of the same color and of opposite colors.


An isomorphic way to define GxH is as the group whose elements are ordered pairs (g, h) where g is in G and h is in H, and whose multiplication is performed pointwise.

i.e. (g,h)(g',h') = (gg',hh')

We can then identify each element g of G with its image (g,1) in GxH. Similarly, we can identify h with (1,h). If we do this, note that the g's commute with the h's under this convention:

(g, 1) (1, h) = (1, h) (g, 1)
 
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  • #3
thanks for that.

But still was a bit confused.But found this in a set of lecture notes on group theory( which I quit):

elements of A x B were written (a,b)
can write this as (a,b) = [itex](a,e_B)(e_A,b)=(e_A,b)(a,e_B)=(ae,eb)=(a,b)[/itex]

shorthand notation for elements of A x B
ab=ba letting [tex](a,e_B) \rightarrow "a" [/itex]
[itex]and (e_A,b) \rightarrow "b"[/tex]
 

1. What is a direct product group?

A direct product group is a mathematical concept that combines two or more groups into a new group. It is denoted by G x H, where G and H are the original groups. The elements of the direct product group are pairs (g,h) where g is from G and h is from H, and the group operation is defined as (g1,h1) * (g2,h2) = (g1 * g2, h1 * h2).

2. How is a direct product group different from a direct sum group?

A direct product group is different from a direct sum group in terms of the group operation. In a direct product group, the group operation is defined as a cartesian product, while in a direct sum group, the group operation is defined as a direct sum. This means that in a direct product group, the elements are pairs, while in a direct sum group, the elements are tuples.

3. What are some examples of direct product groups?

Some examples of direct product groups include the direct product of two cyclic groups, the direct product of two dihedral groups, and the direct product of two symmetric groups. These groups can be represented as G x H, where G and H are the original groups, and the group operation is defined as (g1,h1) * (g2,h2) = (g1 * g2, h1 * h2).

4. What properties do direct product groups have?

Direct product groups have several properties, including being associative, having an identity element, and having inverses. They also inherit the properties of the original groups, such as commutativity and closure. In addition, direct product groups have the property that the order of the group is equal to the product of the orders of the original groups.

5. How are direct product groups used in real-world applications?

Direct product groups have many applications in mathematics and science, such as in group theory, linear algebra, and cryptography. They are also used in computer science to model parallel processing and distributed systems. In chemistry, direct product groups can be used to describe the symmetries of molecular orbitals. Additionally, direct product groups have applications in physics, specifically in the study of quantum mechanics and particle physics.

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