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

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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.

Related facts

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 .