Question on group theory: simplest math construction

In summary, the author of the book of P.Ramond, Group Theory, p8, discusses the use of subgroups in a mathematical construction. Specifically, it is mentioned that D_2 contains three subgroups Z_2, and the question is raised as to why the author only chose to use two of the three subgroups in their construction. It is then questioned what would happen if all three subgroups were used. The author's goal is to create a direct product that is essentially the same group as a 4 element group, but it is noted that a direct product with 8 elements cannot be isomorphic to a group with only 4 elements.
  • #1
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Hello Big Minds,

In the following analysis...It is said that D_2 contains three subgroups Z_2...why did he choose a mathematical constuction contains only two of the the three subgroups? shouldn't he use the three in his construction? what will happen if he used the three?

upload_2015-2-1_10-52-31.png

[from the book of P.Ramond, Group Theory, p8.]

best regards

Abolaban
 
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  • #2
How many distinct elements would the direct product group have if he used 3 subgroups, each with 2 elements in them? If a direct product has 8 elements , it can't be isomorphic to a group with only 4 elements in it. His goal was to write an direct product that is essentially the same group as the 4 element group.
 

1. What is group theory?

Group theory is a branch of mathematics that studies the algebraic structures known as groups. These are sets of elements that satisfy certain mathematical properties and can be combined through operations to form new elements.

2. What is the simplest math construction in group theory?

The simplest math construction in group theory is the cyclic group, which consists of a single element that can be combined with itself through a binary operation to produce new elements.

3. How is group theory used in science?

Group theory has applications in many areas of science, including physics, chemistry, and computer science. It is used to study symmetry and patterns in physical systems, as well as to analyze and solve problems in abstract algebra and geometry.

4. What are some common examples of groups?

Some common examples of groups include the integers under addition, the non-zero real numbers under multiplication, and the symmetries of a regular polygon. Groups can also be defined using matrices, permutations, and other mathematical objects.

5. How does group theory relate to other branches of mathematics?

Group theory has connections to many other branches of mathematics, including number theory, topology, and representation theory. It also has applications in cryptography, coding theory, and other areas of computer science.

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