Is the Group of Order 765 Abelian?

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In summary, to prove that the group of order 765 is abelian, we can use Sylow's third theorem to find that the number of Sylow-3 and Sylow-17 subgroups is 1, making them both normal subgroups. By finding a group homomorphism from G to either subgroup, we can show that G is cyclic. Additionally, a counting argument can be used to show that the p-Sylow subgroups exhaust G, and since each is abelian, G must be abelian as well. This proves that the group of order 765 is abelian.
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Homework Statement


Show that the group of order 765 is abelian (Hint: let G act by conjugation on a normal Sylow p subgroup)


Homework Equations


Sylow theorems


The Attempt at a Solution


By using Sylow`s third theorem, I have calculated that the number of Sylow-3 subgroups and Sylow 17 subgroups is both 1; so both of them are normal subgroups of G. I just don't really understand how to proceed from here; by G acting on either one of those subgroups; I get a group homomorphism G->S_Q (where Q is either the Sylow 17 subgroup or Sylow 3 subgroup). I believe that eventual goal is to show that G is cyclic.

Some random thoughts in my head: Aut(Q)=C_16 (Q being the 17-Sylow subgroup). Much thanks and any help is appreciated
 
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First do a counting argument to show that the number of p-Sylow subgroups for all three values of p must be equal to 1. This will show the p-Sylow subgroups exhaust G. Hence their direct product is isomorphic to G. But each p-Sylow subgroup is abelian (because each order is either prime or a square of a prime) and so all of G is abelian. Done diddly done! You're going to UBC aren't you? :)
 

1. What is a group of order 765?

A group of order 765 refers to a mathematical structure consisting of a set of 765 elements and an operation that combines any two elements in the set to produce a third element. It is a fundamental concept in abstract algebra and has various applications in different fields of science.

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

A group is considered abelian if its operation is commutative, meaning that the order in which the elements are combined does not affect the final result. In other words, if a and b are elements in the group, then a * b = b * a. This property is named after the mathematician Niels Henrik Abel.

3. How do you determine if a group of order 765 is abelian?

To determine if a group of order 765 is abelian, we need to check if its operation satisfies the commutative property. This can be done by considering all possible combinations of elements in the group and verifying if the resulting products are the same regardless of the order in which the elements are combined.

4. What are the possible subgroups of a group of order 765?

A group of order 765 can have various subgroups, including the trivial subgroup {e} consisting of only the identity element, the entire group itself, and subgroups of orders 3, 5, 9, 15, 45, 51, 85, 153, 255, and 765. The number of subgroups of a group of order 765 is related to its prime factorization (3 * 5 * 17).

5. How is the concept of a group of order 765 relevant in science?

The concept of a group of order 765 has various applications in science, particularly in the fields of chemistry, physics, and computer science. In chemistry, for example, it is used to describe the symmetries of molecules and crystals. In physics, it is essential in understanding the symmetries of physical systems and in formulating laws and theories. In computer science, groups are used in cryptography and coding theory.

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