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.
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.