The Sylow theoremsThe Sylow Theorems
Here is my version of the proof of the Sylow theorems. It is the result of taking the proof in Gallian and trying to make it as digestible as possible. In particular, I tried to break the long proof into bite-sized pieces. The main goal here is to convey an overview of how the ingredients fit together, so I'll skip lightly over some of the details.

The prerequisites are basically all of the group theory that came before the Sylow theorems in this course, including: Lagrange's theorem, the first and second isomorphism theorems, and the orbit-stabilizer theorem. I'll also use Cauchy's theorem, even though the book lists it as a corollary to the Sylow theorems (more on that later). I'll assume you know the definition of a Sylow-subgroup and all the terms in the statements of the Sylow theorems. From now on, G is a finite group and p is a prime number that divides the order of G. Recall that a p-subgroup of G is a subgroup of G with order equal to a power of p.

Definition. A maximal p-subgroup of G is a p-subgroup of G that is not contained in any larger p-subgroup of G.
This is only a temporary definition, since it will turn out that a "maximal p-subgroup" is just the same thing as a "Sylow p-subgroup". However it would be circular logic to assume this. Hypothetically, you could imagine a group G of order 100, and a subgroup H of order 2 such that H is not contained in any subgroup of G that has order 4, (or indeed where there is no subgroup of G that has order 4). If there were such an H, it would be maximal but not Sylow. Lemma 1. If H is a maximal p-subgroup of G then the number of conjugates of H is equal to 1 modulo p.

Proof. Let C be the collection of all conjugates of H. Suppose, seeking a contradiction, that the number of elements of C is not equal to 1 modulo p. Let H act on C by conjugation. The size of every orbit must divide the order of H, and hence must be a power of p. Clearly {H} is...

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The Pythagorean Theorem is...

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Pythagoras, the namesake and supposed discoverer of the Pythagorean Theorem, was born on the Greek island of Samos in the early in the late 6th century. Not much is known about his early years of life, however, we do know that Pythagoras traveled through Egypt in the attempt to learn more about mathematics.
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Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two...

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AKSHAY MISHRA
XI A , K V 2 , GWALIOR
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This...

...In mathematics, the Pythagorean theorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can...

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