# Every Sylow subgroup intersects the center nontrivially or is contained in a centralizer

## Statement

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:

1. It intersects the center of  nontrivially.
2. 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.

## Facts used

1. Class equation of a group
2. Sylow subgroups exist
3. Sylow implies order-dominating: Any two Sylow subgroups are conjugate, and any -subgroup is contained in a -Sylow subgroup.
4. Cauchy's theorem for Abelian groups
5. Central implies normal

## Proof

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:

1. Case that  divides the order of :
1. 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 .
2. 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 .
2. Case that  does not divide the order of :
1. 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 .
2.  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 .
3.  contains a subgroup of order : This follows by fact (2).
4. Any -Sylow subgroup of  is of the form  for some : This follows from the previous step, and fact (3).
5. 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 .