The cases in proving that group of order 90 is not simple

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In summary, the problem is split into two cases to exhaust all possibilities. The first case involves counting elements and reaching a contradiction, while the second case considers the intersection of subgroups, which must be a subgroup according to Lagrange's theorem. This leads to the negation of two distinct Sylow 3-subgroups intersecting trivially, and instead intersecting with 3 elements. This is explained further in pages 65-66 of the class notes provided.
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Mr Davis 97
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I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that let's us count elements and get a contradiction, but why is the second case there? In other words, why do these two cases exhaust all possibilities?

EDIT: Actually, I think that I see it now. Since the intersection of subgroups is a subgroup, by Lagrange we must have that the negation of two distinct Sylow 3-subgroups intersecting trivially is intersecting with 3 elements.
 
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see pages 65-66 of these class notes:

http://alpha.math.uga.edu/%7Eroy/843-1.pdf
 

Related to The cases in proving that group of order 90 is not simple

1. What is a group of order 90?

A group of order 90 is a mathematical concept that refers to a group with 90 distinct elements. A group is a set of objects that follow specific rules for combining them together, such as addition or multiplication. The order of a group is the number of elements it contains.

2. What does it mean for a group to be simple?

A simple group is a group that does not have any non-trivial normal subgroups, meaning there are no smaller groups within it that can be created by taking a subset of its elements. In other words, a simple group cannot be broken down into smaller groups.

3. Why is it important to prove that a group of order 90 is not simple?

Proving that a group of order 90 is not simple is important because it helps to understand the structure of the group and its subgroups. It also has implications in other areas of mathematics, such as algebra and number theory.

4. How do you prove that a group of order 90 is not simple?

There are several methods that can be used to prove that a group of order 90 is not simple. One approach is to show that the group has non-trivial normal subgroups, meaning there are smaller groups within it that can be created by taking a subset of its elements. Another method is to use the Sylow theorems, which provide a way to calculate the number of subgroups of a given order in a group.

5. What are some real-world applications of proving that a group of order 90 is not simple?

The concept of groups and their properties have applications in various fields, including cryptography, physics, and chemistry. Proving that a group of order 90 is not simple can help in solving computational problems and understanding the structure of complex systems. It also has implications in understanding the symmetries and patterns in nature.

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