Abstract Algebra - Direct Product Question

In summary, the conversation discusses finding a non-trivial group G that is isomorphic to G x G. It is determined that G must be infinite, and the group Z is proposed as a potential solution. However, it is pointed out that Z x Z is not isomorphic to Z and is not cyclic. The idea of using the group of integer functions on the integers is suggested. It is also mentioned that the map described by the original solution needs to be proven to be a homomorphism and have an inverse. The conversation then moves on to discussing the number of generators needed to generate GxG, which is determined to be at least 2n if G is finitely generated.
  • #1
BSMSMSTMSPHD
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I'm supposed to find a non-trivial group G such that G is isomorphic to G x G.

I know G must be infinite, since if G had order n, then G x G would have order n^2. So, after some thought, I came up with the following. Z is isomorphic to Z x Z.

My reasoning is similar to the oft-seen proof that the rationals are countable.

Picture a grid with dots representing each element in Z x Z. Now, starting at the origin, trace a circuitous path (in any direction, but always a tight spiral) and define a map that sends 0 to (0,0), 1 to the next point, -1 to the next point, 2 to the next point, etc.

Is it enough to describe this map in the way I have, or do I need further information (or am I wrong?)

Thanks.
 
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  • #2
Z x Z is not isomorphic to Z. Z x Z is not cyclic. You might think about the group of integer functions on the integers.
 
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  • #3
Is it enough to describe this map in the way I have, or do I need further information (or am I wrong?)
That describes a map... (although having only a heuristic description makes it hard to prove things about it)

But you've yet to prove that your map is a homomorphism, that it has an inverse, and that its inverse is a homomorphism.
 
  • #4
Thanks guys. Back to the drawingboard.
 
  • #5
Hrm. I hate to give big hints like this, but...

If a generating set of G must contain at least n elements... then (heuristically speaking) how many elements must a generating set of GxG contain?
 
  • #6
I don't like to give answers like this, but I haven't the slightest idea.
 
  • #7
Well, how many generators does it take to generate the subgroup Gx1 of GxG?
 
  • #8
I would say n - same as for G.
 
  • #9
And what about 1xG? So what does that suggest will be (roughly) true, if you want to generate all of GxG?
 
  • #10
You would need 2n?
 
  • #11
Right. In particular, if G is finitely generated, then...

(this is not a rigorous proof -- I don't know if weird things will happen that allow you to use less than 2n... but we're not looking for proofs here, we're searching for examples!)
 

Related to Abstract Algebra - Direct Product Question

1. What is the direct product in abstract algebra?

The direct product in abstract algebra is a binary operation that combines two algebraic structures into a new structure, where the elements of the new structure consist of ordered pairs of elements from the original structures. It is denoted by the symbol × and is sometimes referred to as the Cartesian product.

2. How is the direct product different from the direct sum?

The direct product and the direct sum are both binary operations that combine two algebraic structures. However, the direct sum only combines structures of the same type, while the direct product can combine structures of different types. Additionally, the direct sum preserves the original structures, while the direct product creates a new structure.

3. What is an example of a direct product in abstract algebra?

An example of a direct product in abstract algebra is the direct product of two groups. If we have two groups G and H, the direct product of G and H would be the set of all ordered pairs (g,h) where g is an element of G and h is an element of H. The direct product operation would then be defined as (g1,h1) × (g2,h2) = (g1g2, h1h2).

4. How is the direct product related to the direct sum?

The direct product and the direct sum are related in that they both involve combining algebraic structures. In fact, the direct product can be seen as a generalization of the direct sum, as it allows for the combination of structures of different types. Additionally, the direct sum can be seen as a special case of the direct product, where the structures being combined are identical.

5. What are some applications of the direct product in abstract algebra?

The direct product has numerous applications in mathematics and other fields. In abstract algebra, it is used to create new structures from existing ones, allowing for the study of more complex systems. It is also used in coding theory and cryptography, as well as in physics and chemistry to model the behavior of particles and molecules. Additionally, the direct product has applications in computer science, particularly in the design of databases and data structures.

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