# Applications of Sylow theorems [duplicate]

Almost all the books of algebra or group theory give following types of applications of Sylow theorems:

A group of order $...$ is not simple.

A group of order ... has normal Sylow subgroup.

Even in math.stackexchange, there are many questions with above title and they involve questions which are related to proving above type of statements.

Are there any other type of applications of these theorems of Sylow?

## marked as duplicate by Trevor Gunn, Daniel W. Farlow, Leucippus, JonMark Perry, José Carlos SantosJun 29 '17 at 6:33

The revised version of Hartshone's "Elementary Projective Geometry" contains a nice proof, based on the Sylow Theorem, that the structure of a certain group of order $168$ implicitly determines the "geometry" of the 7-point projective plane (i.e., that certain conjugacy classes correspond to "points" and others to "lines", etc.) I'm not certain this is what you want, and to get to this, there are many times when he says things like "and so the 7-sylow subgroup is unique", but that's no surprise --- counting the number of $p$-Sylow subgroups is what the Theorems let you do!

There are many more applications of the Sylow Theorems. Here is a small list:

$\bullet$ Classification of groups of order $pq$, for $p<q$ primes.

$\bullet$ A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups.

$\bullet$ Every finite $p$-group is isomorphic to some subgroup of the upper unitriangular group.

$\bullet$ The Schur-Zassenhaus-Theorem: Every short exact sequence of finite groups $$1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$$ with $gcd(N|,|Q|)=1$ splits.

Several other applications are given at this MO-question.