Isomorphism from group to a product group

In summary, the conversation discussed determining whether G is isomorphic to the product group HXK, where G=ℂx, H is the unit circle, and K is the set of positive real numbers. The conversation also mentioned the multiplication map and its image, as well as the conditions for an isomorphism between HXK and G. The question was raised about viewing the unit circle as either a set of points or as cos(θ)+i*sin(θ), and a solution was proposed using bijection and preservation of operations.
  • #1
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Homework Statement



Determine whether or not G is isomorphic to the product group HXK.

G=ℂx
H={unit circle}
K={Positive real numbers}



Homework Equations



Let H and K be subroups of G, and let f:HXK→G be the multiplication map, defined by f(h,k)=hk. Its image is the set HK={hk: h in H, k in K}.

f is an isomorphism from the product group HXK to G iff H intersect K is the identity,HK=G, and also H and K are normal subgroups of G.


The Attempt at a Solution



My actual question is how am I supposed to look at the unit circle?
I know if I view it as all the points (a,b) on the unit circle then its intersection with K would be the empty set and thus G wouldn't be isomorphic to HXK.

but I have a feeling that is too simple. Am I potentially supposed to view the unit circle as cos(θ)+i*sin(θ) where θ is in ℝ?
 
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  • #2
Any complex number in ℂx can be represented as re and the reverse holds as well.
So z → (r,θ) is a bijection.
If you define the operations in HxK to be regular multiplication, respectively regular addition, you have preservation of the operation, since it literally is the same.
 

1. What is isomorphism from group to a product group?

Isomorphism from group to a product group is a mathematical concept that refers to the relationship between two groups where one group can be mapped onto the other group in a way that preserves the group structure, meaning the operation and identity elements are preserved.

2. How is isomorphism different from homomorphism?

Isomorphism and homomorphism are both mathematical concepts that describe the relationship between two groups. However, isomorphism is a stronger condition than homomorphism, as it requires not only that the group structure is preserved, but also that the mapping is bijective, meaning it is both one-to-one and onto.

3. What is the significance of isomorphism in group theory?

Isomorphism is an important concept in group theory because it allows us to identify groups that have the same structure, even if the elements of the groups may look different. This allows us to study one group and apply our knowledge to another group with the same structure, making it a powerful tool for understanding groups.

4. Can all groups be isomorphic to a product group?

No, not all groups can be isomorphic to a product group. In order for two groups to be isomorphic, they must have the same number of elements and the same group structure. This means that groups with different structures, such as cyclic and non-cyclic groups, cannot be isomorphic to a product group.

5. What is an example of isomorphism from a group to a product group?

An example of isomorphism from a group to a product group is the isomorphism between the additive group of integers (Z) and the product group of integers and 2-element group (Z x Z/2Z). This isomorphism is given by the map f: Z -> Z x Z/2Z, where f(n) = (n, n mod 2).

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