Abstract Algebra - Subgroups of index 2 in R*

In summary, a subgroup of index 2 in R* is a subset of the multiplicative group of nonzero real numbers that contains exactly half of the elements of R*. It can be identified by its defining property of having an element in the subgroup and an element not in the subgroup whose product is in the subgroup. These subgroups are significant in abstract algebra and have applications in various mathematical fields. They can only contain real numbers and are closely related to other subgroups in R*.
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Homework Statement



I am having trouble understanding how I would go about finding all subgroups of index 2 in R*, the multiplicative group of nonzero real numbers. Any hints will be greatly appreciated.

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The Attempt at a Solution


 
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If you don't know what you need to do, then look for something you can do.

For example, you spent a lot of time dealing with structure theorems for Abelian groups -- so one of the most natural things to do when studying an Abelian group is to see if you can analyze its structure...
 

1. What is a subgroup of index 2 in R*?

A subgroup of index 2 in R* is a subset of the multiplicative group of nonzero real numbers that contains exactly half of the elements of R*. This means that for every element in the subgroup, there is another element in R* that is not in the subgroup.

2. How can a subgroup of index 2 in R* be identified?

A subgroup of index 2 in R* can be identified by finding its defining property, which is that for every element a in the subgroup, there is another element b in R* that is not in the subgroup, but whose product ab is in the subgroup.

3. What is the significance of subgroups of index 2 in R*?

Subgroups of index 2 in R* are important in the study of abstract algebra because they represent a particular type of symmetry in the group structure. They also have applications in various mathematical fields, such as number theory and geometry.

4. Can subgroups of index 2 in R* have elements other than real numbers?

No, subgroups of index 2 in R* can only contain real numbers because R* is defined as the set of nonzero real numbers. If a subgroup contains any other type of element, it would not be a subgroup of R*.

5. How are subgroups of index 2 in R* related to other subgroups?

Subgroups of index 2 in R* are closely related to other subgroups because they can be used to construct other subgroups. For example, if a subgroup of index 2 in R* is combined with its complement, a subgroup of index 2 is formed. They can also be used to classify and study other types of subgroups in R*.

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