Every Sylow subgroup intersects the center nontrivially or is contained in a centralizer
Suppose is a finite group and is a prime dividing the order of . Then, every -Sylow subgroup of satisfies at least one of these two conditions:
- It intersects the center of nontrivially.
- it is contained in the centralizer of a non-central element.
Further, if any one -Sylow subgroup satisfies a particular condition, so do all the others.
- Sylow subgroups exist: In fact, one of the proofs of the existence of Sylow subgroups essentially uses this idea.
- Sylow implies order-conjugate: Any two -Sylow subgroups are conjugate.
- Class equation of a group
- Sylow subgroups exist
- Sylow implies order-dominating: Any two Sylow subgroups are conjugate, and any -subgroup is contained in a -Sylow subgroup.
- Cauchy's theorem for Abelian groups
- Central implies normal
Given: A finite group of order , where is prime, is a positive integer, and does not divide .
To prove: Every -Sylow subgroup of either intersects the center nontrivially, or is contained in the centralizer of a non-central element.
Proof: Consider the class equation of (fact (1)):
where are the conjugacy classes of non-central elements and is an element of for each .
We consider two cases:
- Case that divides the order of :
- There exists a normal subgroup of order in : Since is Abelian, fact (4) yields that it has a subgroup of order . Since is in the center, is normal in (by fact (5)). Thus, is a normal subgroup of of order .
- Suppose is any -Sylow subgroup of . By fact (3), the subgroup is contained in some conjugate of . Since is normal, this forces . Thus, intersects the center nontrivially -- the intersection contains a subgroup of order .
- Case that does not divide the order of :
- There exists such that does not divide the index of in : Since divides the order of , cannot divide the index of every , otherwise the class equation would yield that divides the order of .
- is a proper subgroup of whose order is a multiple of : Since is non-central, is proper in . Further, since is relatively prime to , Lagrange's theorem (fact (3)) yields that the order of is , which is a multiple of .
- contains a subgroup of order : This follows by fact (2).
- Any -Sylow subgroup of is of the form for some : This follows from the previous step, and fact (3).
- Any -Sylow subgroup of is contained in the centralizer of a non-central element: This follows from the previous step; in fact, it is contained in .