Application of Sylow's theorems

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In summary, in an application of Sylow's theorems, we have a group of order 28 = 2^2 x 7 and either 1 or 7 sylow 2-subgroups. If we assume 7 sylow 2-subgroups, we have 21 non-identity elements and 1 sylow 7-subgroup. The question of why the 7 sylow 2-subgroups have trivial intersection is not clearly explained, but it is important to note that this is not always the case for Sylow subgroups. However, if there are 7 sylow 2-subgroups, their union can have at most 21 non-identity elements and the
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I saw in an application of Sylow's theorems, it said we have something like a group of order 28 = 2^2 x 7, so we have either 1 or 7 sylow 2-subgroup. Assuming we have 7 sylow 2-subgroups, then we have 21 non-identity elements and the identity, and we have 1 sylow 7-subgroup, blah blah blah...

the point I wanted to know that was never really explained clearly was that why are the 7 sylow 2-subgroups intersection trivial? I realize that they are conjugates of each other, but this counting of elements seems kind of moot unless we know that these subgroups don't have any common elements.
 
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Sylow subgroups do NOT in general have trivial intersection. What you can say, if you have 7 Sylow 2-subgroups, is that their union is at MOST 21 non-identity elements plus the identity.
 

1. What is Sylow's theorem and how is it applied in scientific research?

Sylow's theorem is a mathematical concept that is used to analyze the structure of groups. It states that if a group has a prime power order, then it must have a subgroup of that order. In scientific research, Sylow's theorem can be applied in various fields such as physics, chemistry, and biology to understand the underlying structure of molecules, particles, and organisms.

2. How does the application of Sylow's theorem help in understanding the symmetry of a system?

Sylow's theorem can be used to determine the possible subgroups of a group, which in turn, helps in identifying the symmetries present in a system. This can be useful in studying the properties of crystals, molecules, and other complex systems where symmetry plays a crucial role in their behavior.

3. Can Sylow's theorem be applied in cryptography and data security?

Yes, Sylow's theorem has applications in cryptography and data security. It can be used to analyze the structure of certain mathematical groups that are used in encryption algorithms. This helps in understanding the strength of these algorithms and identifying potential vulnerabilities.

4. How is Sylow's theorem used in group theory and abstract algebra?

Sylow's theorem is a fundamental concept in group theory and abstract algebra. It is often used to prove other theorems and to understand the structure of different types of groups. It also has applications in other areas of mathematics such as number theory and combinatorics.

5. What are some real-life examples of the application of Sylow's theorem?

Sylow's theorem has numerous real-life applications, such as in crystallography to understand the symmetry of crystals, in genetics to study the inheritance patterns of certain traits, and in chemistry to analyze the structure of molecules. It is also used in various fields of physics, including quantum mechanics, particle physics, and cosmology.

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