Findng a group between two groups

In summary: But here is another example: consider the set of all positive real numbers, and define the group operation to be exponentiation. This forms a group which is a proper subgroup of the reals under multiplication.
  • #1
anugrah
2
0
we all know that set of rationals i a subgroup of set of reals. my question is whether there exsts a group between these tw groups. f yes what it can be? and if no, how to prve the non-existence?
 
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  • #2
There exist uncountably many. (We assume that you mean addition as the group law.)
 
  • #3
It will help your intuition if you notice that the reals form a 'very large' vector space over the rationals.
 
  • #4
sorry, forgot to mention that the group law is multiplication. I wanted everything with respect to multiplication. And if there exists a subgroup between these two groups (even with respect to addition ) please give an example of the same.
 
  • #5
anugrah said:
sorry, forgot to mention that the group law is multiplication. I wanted everything with respect to multiplication. And if there exists a subgroup between these two groups (even with respect to addition ) please give an example of the same.

Pick any irrational number k and consider the set Q'={qkn | q is a nonzero rational number, and n is an integer}

It is easy to show that this is a group under multiplication, and easy to show that the nonzero rationals are a subgroup of this group, but clearly since this group is countable it is proper subgroup of the reals.
 
  • #6
anugrah said:
sorry, forgot to mention that the group law is multiplication.

The real numbers under multiplication are not a group.


I wanted everything with respect to multiplication. And if there exists a subgroup between these two groups (even with respect to addition ) please give an example of the same.

It would be better if you tried to work it out yourself.
 

Related to Findng a group between two groups

1. How do you determine which group is in between two other groups?

The most common way to determine which group is in between two other groups is by conducting a statistical analysis. This involves collecting data from both groups and using mathematical calculations to determine the average and standard deviation. The group with the average closest to the midpoint between the averages of the other two groups is considered to be in between them.

2. Can there be more than one group in between two groups?

Yes, it is possible for there to be multiple groups in between two other groups. This can happen if there are several distinct subgroups within the overall population that fall within the range of the two groups being compared.

3. What are some other methods for finding a group between two groups?

Aside from statistical analysis, other methods for finding a group between two groups include conducting surveys or interviews to gather qualitative data, observing behavior or attitudes within the different groups, or using a combination of both quantitative and qualitative methods.

4. How do you ensure the accuracy of the results when finding a group between two groups?

To ensure the accuracy of the results, it is important to use a large and diverse sample size, collect data from multiple sources, and use reliable and validated measurement tools. It is also crucial to properly analyze and interpret the data and consider any potential biases or limitations in the research methodology.

5. What are some potential applications of finding a group between two groups?

Finding a group between two groups can have various applications in different fields such as market research, social sciences, and healthcare. For example, it can help identify target audiences for a product or service, understand cultural or societal differences, or identify risk factors for certain health conditions.

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