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5: The Sylow Theorems; Application

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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.


This page titled 5: The Sylow Theorems; Application is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James S. Milne.

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