## 10 Jan Applications of Sylow Theorems

**Sylow Theorem’s** is Group Theory are an important mathematical tool. Below we give a simple application of the theorems.

**Theorem:** If and are subgroups of and , then is a subgroup of . In particular, if is normal is then for any .

**Proof:** We prove . We let .

By assumption, , hence .

This proves that .

Similarly, , proving the reverse.

We know that if and are subgroups of a group then is a subgroup if and if only .

This proves the result.

**Theorem:** Let be a group of order , where and are primes such that .

- If , there exists a non-abelian group of order .
- Any two non-abelian groups of order are isomorphic.

**Proof:** We let be a Sylow -subgroup of the symmetric group of degree , . We know that if is a prime and is a subgroup of of order , then .

We know that every conjugate of contains exactly -cycles and computing the index of in we can prove the above result.

So, .

Since and by Cauchy’s theorem has a subgroup of order .

Using the previous theorem we can see that is a group of order .

Since so, is non-abelian.

This proves the first result.

Let be any group of order , let and let .

We have and let .

Since is cyclic, it contains a unique subgroup of order , say , and any homomorphism must map to a power of .

There are therefore homomorphisms given by .

Now each for not equal to give rise to a non-abelian group , of order .

It is straightforward to check that these groups are all isomorphic because for each , there is some generator of such that .

Thus up to a choice for the arbitary generator of , these semi-direct products are all the same.

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