5: The Sylow Theorems; Application
( \newcommand{\kernel}{\mathrm{null}\,}\)
In this chapter, all groups are finite.
Let G be a group and let p be a prime dividing (G:1). A subgroup of G is called a Sylow p-subgroup of G if its order is the highest power of p dividing (G:1). In other words, H is a Sylow p-subgroup of G if it is a p-group and its index in G is prime to p.
The Sylow theorems state that there exist Sylow p-subgroups for all primes p dividing (G:1), that the Sylow p-subgroups for a fixed p are conjugate, and that every p-subgroup of G is contained in such a subgroup; moreover, the theorems restrict the possible number of Sylow p-subgroups in G.