Showing that two groups are not isomorphic question

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In summary, the conversation discusses how to show that ##\mathbb{R} - \{ 0\}## is not isomorphic to ##\mathbb{C} - \{0 \}##. One way is to look at the solutions to ##x^3 = 1##, which differ in the two sets. The use of ##x^2 = -1## is less clear, but one idea is to consider elements of finite order, where there are only two in ##\mathbb{R}-\{0\}## and infinitely many in ##\mathbb{C}-\{0\}##. However, using the single relation ##\varphi(r)=i## can lead
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Mr Davis 97
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I am trying to show that ##\mathbb{R} - \{ 0\}## is not isomorphic to ##\mathbb{C} - \{0 \}##. If we simply look at ##x^3 = 1##, it's clear that ##\mathbb{R} - \{ 0\}## has one solution while ##\mathbb{C} - \{0 \}## has three.

My question, how can I use ##x^2 = -1## to show that they are not isomorphic? Using ##x^3 = 1## is more clear because any isomorphism would preserve powers and preserve the identity. But using ##x^2 = -1## is less clear to me.
 
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Interesting question. I would consider all elements of finite order. There are only two of them in ##\mathbb{R}-\{0\}## and infinitely many in ##\mathbb{C}-\{0\}##. But if you only want to use the single relation ##\varphi(r)=i## it's a bit tricky, because one easily falls into unproven statements like the ordering of the two sets and an assumed isomorphism isn't necessarily order preserving. Do you have any ideas?

Edit: I think I got it: Consider ##\varphi(-1)\cdot \varphi (-1)##.
 
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1. How do you show that two groups are not isomorphic?

To show that two groups are not isomorphic, you must prove that there is no bijective homomorphism between the two groups. This can be done by examining the groups' structures, such as their order, subgroups, and elements, and showing that they do not have the same properties.

2. Can two groups with the same order be isomorphic?

Yes, two groups with the same order can be isomorphic. However, it is not a guarantee. Isomorphism depends on the internal structure of the groups, not just their order. Therefore, it is important to examine other properties of the groups to determine if they are isomorphic or not.

3. What are some common techniques used to show that two groups are not isomorphic?

One common technique is to look at the groups' subgroups and determine if they have the same number of subgroups of each order. Another technique is to compare the groups' elements and see if they have the same number of elements of each order. Additionally, examining the groups' cyclic subgroups and their generators can also be helpful.

4. Is it possible for two groups to be isomorphic in more than one way?

No, two groups can only be isomorphic in one way. Isomorphism is a one-to-one correspondence between groups that preserves their structures. If two groups were isomorphic in more than one way, it would mean that there are multiple ways to map the elements of one group to the other while preserving their structures, which is not possible.

5. Are there any shortcuts or tricks to quickly show that two groups are not isomorphic?

Unfortunately, there are no shortcuts or tricks to quickly show that two groups are not isomorphic. It requires careful examination and analysis of the groups' structures. However, having a good understanding of the properties and characteristics of different groups can make the process faster and more efficient.

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