5: The Sylow Theorems; Application
- Page ID
- 180009
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In this chapter, all groups are finite.
Let \(G\) be a group and let \(p\) be a prime dividing \((G\colon1)\). 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 \((G\colon1)\), 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 \(G\).


